Storm Backgrounds

Four storms have complete dataset: Milton, Helene, Idalia & Ian

Figure: TC Tracks, Spotter Trajectories and Model Domain Maps

Basis of Comparison - Data

Buoys

Wave observations were obtained from Sofar Spotter buoys. Spotter products are derived from spectral estimates calculated over a finite observation window and therefore represent time-averaged sea-state conditions rather than instantaneous measurements. Bulk wave statistics and directional spectra reported at a given timestamp are representative of the surrounding averaging interval (typically 30 minutes), reflecting the integrated wave response to recent wind forcing and wave evolution processes.

The buoy measures: $\overset{ˉ}{U}_{o b s} (t) = \frac{1}{Δ t} \int_{t - Δ t /2}^{t + Δ t /2} U (τ) d τ$ with $Δ t = 30 min$

COAMPS-TC Model

Atmospheric and oceanic forcing fields were obtained from the Coupled Ocean–Atmosphere Mesoscale Prediction System for Tropical Cyclones (COAMPS-TC). Model output consists of gridded, instantaneous realizations of the simulated atmosphere–ocean state at discrete output times, including 10-m wind speed and direction. Unlike HURDAT2 intensity estimates or temporally averaged buoy observations, COAMPS-TC fields represent instantaneous model conditions and are used here to characterize the local storm environment and wind forcing experienced by the observed wave field.

Typically an instantaneous value at output times : $U_{m o d e l} (t_{n})$ .

COAMPS-TC output was analyzed using the unadjusted (100%) 10-m wind fields. In addition to the raw model winds, COAMPS-TC provides an operationally adjusted 89% wind product in which wind speeds are reduced by a factor of 0.89. This adjustment was developed to improve agreement between modeled maximum winds and operational tropical cyclone intensity estimates, such as those reported in best-track archives. Because the present study focuses on local wind forcing and wave evolution rather than storm intensity verification, the unadjusted wind fields were used to more directly represent the modeled atmosphere–ocean environment experienced by the observed wave field.

HURDAT2 Best Track Data

Tropical cyclone track and intensity information were obtained from the HURDAT2 best-track archive. HURDAT2 provides operational post-storm estimates of storm-center position and maximum sustained 10-m wind speed, reported as a 1-minute sustained wind averaged over the storm’s region of maximum winds. Best-track positions and intensities are provided at 6-hour intervals and represent a storm-scale characterization of cyclone evolution rather than local environmental conditions at a specific observation point.

Consequently-

The three datasets describe different aspects of the tropical cyclone system: HURDAT2 provides storm-scale operational intensity estimates, Spotter buoys provide temporally averaged local wave observations, and COAMPS-TC provides instantaneous gridded representations of the modeled atmosphere–ocean environment. This distinction is important when interpreting comparisons between observed and modeled quantities.

Model output was available at 15-minute intervals for Helene, Milton, and Idalia and at 60-minute intervals for Ian. To maintain consistency across storms, environmental quantities were obtained by temporally interpolating model fields to the buoy observation times. This approach assumes that storm evolution between model output times is smooth relative to the temporal sampling interval.

Approach to compare similar quantities across datasets

COAMPS-TC fields were collocated with buoy observations using different temporal procedures depending on model output frequency. For storms with 15-minute model output, fields were first linearly interpolated to the buoy position and then averaged over the three model times centered on the buoy timestamp, approximating the 30-minute Spotter observation window: $\overline{X}_{m o d e l} (t_{b}) = \frac{1}{3} [X (t_{b} - 15 min) + X (t_{b}) + X (t_{b} + 15 min)]$ , where $X$ represents a collocated model variable and $t_{b}$ is the buoy timestamp.

For the storm with 60-minute model output, fields were linearly interpolated in time to $t_{b}$ : $X_{m o d e l} (t_{b}) = X (t_{i}) + \frac{t _{b} - t _{i}}{t _{i + 1} - t _{i}} [X (t_{i + 1}) - X (t_{i})]$

The 15-minute averaging procedure better matches the finite Spotter sampling window, while the 60-minute interpolation provides the best available estimate of the local model state but cannot resolve sub-hourly variability. Therefore, collocation uncertainty is expected to be larger for the 60-minute storm, especially near strong gradients in wind speed, storm radius, or eyewall structure. The persistence of coherent storm-relative patterns across both 15-minute and 60-minute model resolutions suggests that the primary relationships identified here are not solely artifacts of model output frequency, although resolution-dependent uncertainty should be considered when interpreting storm-specific differences.

Tracking the TC center

Tropical cyclone centers were identified using a composite cost function incorporating sea-level pressure, low-level relative vorticity, near-eye wind speed, and temporal track continuity. This approach was motivated by previous work demonstrating that tropical cyclone center location is not uniquely defined by a single field extremum, as the minimum pressure, maximum vorticity, and circulation center may become spatially separated in asymmetric or evolving storms. Rather than identifying the center solely as the point of minimum pressure, the cost function was designed to locate the dynamically coherent vortex center by favoring regions of low pressure, enhanced cyclonic vorticity, weak near-eye winds, and physically realistic storm motion between successive time steps.

The composite approach is consistent with the view that the tropical cyclone center is more appropriately interpreted as the center of the symmetric vortex structure rather than the location of any single pressure or wind extremum. By combining multiple dynamical fields and enforcing temporal continuity, the resulting trajectory is less sensitive to transient asymmetries, eyewall structure, or localized model noise while retaining the large-scale evolution of the storm circulation.

Cost Function

$J (x, y, t) = w_{p} \overset{p}{^}^{'} (x, y, t) - w_{ζ} \hat{ζ} (x, y, t) + w_{u} \hat{U}_{ey e} (x, y, t) + w_{a} \hat{A} (x, y, t) + w_{s} \hat{D} (x, y, t)$

Where the TC center is chosen as: $(x_{c}, y_{c})_{t} = ar g min_{x, y} J (x, y, t)$

$\overset{p}{^}^{'} (x, y, t)$ -> normalized sea-level pressure anomaly. Lower pressure is favored. $\hat{ζ} (x, y, t)$ -> normalized low-level cyclonic relative vorticity, often at 850 hPa or the lowest usable model level. The minus sign means larger cyclonic vorticity lowers the cost. $\hat{U}_{ey e} (x, y, t)$ -> normalized near-eye 10-m wind speed. Lower wind speed near the candidate center is favored, helping identify the calm circulation center rather than the eyewall. $\hat{A} (x, y, t)$ -> azimuthal asymmetry or structural misfit around the candidate center. Lower values indicate the storm looks more axisymmetric around that point. for example : $A (x, y, t) = \frac{\sum _{r} \sum _{θ} [ q ( r , θ , t ) - q ( r , t ) ] ^{2}}{\sum _{r} \sum _{θ} q ( r , θ , t ) ^{2}}$ where q could be pressure anomaly, tangential wind, or vorticity sampled in an annulus around the candidate center. $\hat{D} (x, y, t)$ -> track-continuity penalty, for example: $D (x, y, t) = ∥ (x, y) - (x_{c}, y_{c})_{t - Δ t} ∥^{2}$ this discourages unrealistic jumps between consecutive time steps.

The weights $w_{p}, w_{ζ}, w_{u}, w_{a}, w_{s}$ control the relative importance of pressure, vorticity, eye wind minimum, azimuthal symmetry, and trajectory smoothness

Tracking Validation Suite (theory)

The validation suite evaluates whether the composite COAMPS-TC center tracker identifies a physically coherent tropical cyclone vortex center suitable for storm-relative analyses. Rather than relying on a single field extremum, the tracker combines information from sea-level pressure, cyclonic vorticity, near-eye wind speed, vortex symmetry, and temporal continuity through a composite cost function: $J (x, y, t) = w_{p} \overset{p}{^} - w_{ζ} \hat{ζ} + w_{u} \hat{U}_{ey e} + w_{a} \hat{A} + w_{s} \hat{D}$ where the selected center is: $(x_{c}, y_{c})_{t} = ar g min J (x, y, t)$ All cost-function components are robustly normalized within the local search window using median and interquartile range scaling. The validation framework assesses the robustness, uniqueness, physical realism, and scientific sensitivity of the resulting center trajectories.

1) Candidate center consistency

Goal: Determine whether independent center definitions identify the same vortex location.

Figure 01 compares trajectories derived from:

Minimum sea-level pressure
Maximum cyclonic vorticity
Minimum near-eye wind speed
Minimum azimuthal asymmetry
Composite center tracker
Interpolated HURDAT2 center using both map-view tracks and latitude/longitude time series.

Agreement among candidate centers indicates a well-defined vortex center. Divergence between candidate centers suggests structural asymmetry, competing vortex signatures, or sensitivity to individual fields.

2) Cost function behavior and center uncertainty

Goal: Evaluate how the composite tracker selects centers and quantify ambiguity in the cost landscape.

Figure 02 displays contributions from : $w_{p} \overset{p}{^}, - w_{ζ} \hat{ζ}, w_{u} \hat{U}_{ey e}, w_{a} \hat{A}, w_{s} \hat{D}$ along with total $J_{min}$ to identify which terms dominate center selection.

Figure 05 evaluates center uniqueness using: $Δ J = J_{2 n d} - J_{1 s t}$ through both time-series and histogram views.

Figure 06a maps the local cost field $J (x, y)$ for selected times, showing the composite and HURDAT2 center. Large $Δ J$ and a single deep minimum indicate a robust center estimate. Small $Δ J$ or multiple comparable minima indicate ambiguity and potential center hopping.

3) Dynamical plausibility of center motion

Goal: Determine whether the diagnosed trajectory exhibits physically realistic tropical cyclone motion.

Metrics: Translation speed: $V_{c} = \frac{d ( x _{t} , x _{t - 1} )}{Δ t}$ Acceleration: $a_{c} = \frac{V _{c} ( t ) - V _{c} ( t - Δ t )}{Δ t}$ Curvature: $κ = \frac{∣ x ^{'} y ^{''} - y ^{'} x ^{''} ∣}{( x ^{'2} + y ^{'2} ) ^{3/2}}$

Figure 03 compares raw and smoothed trajectories for translation speed, acceleration and curvature. Impulsive speed, acceleration, or curvature spikes often indicate center hopping between competing minima rather than true storm motion.

4) Physical consistency with storm structure

Goal: Assess whether the diagnosed center remains physically tied to the modeled cyclone circulation.

Figure 06b maps local SLP with tracked and HURDAT2 centers overlaid. Evaluates whether the center remains within the pressure bowl.

Figure 06c maps cyclonic relative vorticity: $ζ = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$ with tracked and HURDAT2 centers overlaid. Evaluates whether multiple vorticity lobes are competing for center selection.

Figure 04 examines distance to HURDAT2: $Δ r = d (x_{C O A M P S}, x_{H U R D A T 2})$ through both time series and distribution statistics (median, 95th percentile, max distance). This comparison provides a sanity check against the observed best-track position while recognizing that the objective model-space center and operational best-track center need not coincide exactly.

5) Scientific sensitivity for buoy collocation

Goal: Determine whether center-tracking uncertainty materially affects storm-relative buoy coordinates and subsequent wave analyses.

Storm relative coordinates, radius: $r = x^{2} + y^{2}$ and azimuth: $θ = tan^{- 1} (\frac{right}{forward})$ where coordinates are defined relative to storm motion.

Figure 07 compares buoy radius and azimuth using a) HURDAT2 centers, b) raw composite centers and c) smoothed composite centers. Metrics: $r_{r a w} - r_{H U R D A T 2}$ $r_{s m oo t h} - r_{H U R D A T 2}$ $θ_{r a w} - θ_{H U R D A T 2}$ $θ_{s m oo t h} - θ_{H U R D A T 2}$

Figure 08 assesses quadrant assignment stability via confusion matrices comparing HURDAT2 vs raw centers and HURDAT2 vs smoothed centers for all quadrants.

Figure 09 examines radius assignment stability via $r_{r a w} - r_{s m oo t h}$ through time series and histogram analyses.

Figure 10 explores smoothing sensitivity by comparing storm-relative coordinates derived from: raw trajectory, rolling median, Savitzky-Golay, and Spline smoothing. These diagnostics directly assess whether center-tracking uncertainty influences the physical quantities used in the wave analysis.

6) Parameter sensitivity experiments

Goal: determine whether the diagnosed center trajectory is robust to reasonable algorithm choices

Figure 11 explores search radius sensitivity by rerunning the tracker with varying search radii (50, 75, 100, 150km) and compares resulting tracks

Figure 12 explores weight sensitivity by rerunning the tracker using varying parameter weights: Baseline weighting Pressure-dominant weighting Vorticity-dominant weighting Symmetry-dominant weighting to assess sensitivity to the relative importance of individual cost-function components.

A robust vortex center should exhibit similar large-scale motion across reasonable search-radius and weighting choices. Strong divergence indicates sensitivity to local minima or specific cost-function terms.

(1) Is there a unique center? → (2) How is the center selected? → (3) Is the trajectory physically realistic? → (4) Does it correspond to the storm structure? → (5) Does it matter for the buoy science? → (6) Is it robust to methodological choices?

Tracking Validation Suite (output)

Summary figures describe a diagnostic framework for understanding when the COAMPS-TC center estimate is physically trustworthy, why it is trustworthy and how uncertainty propagates into the storm-relative wave analysis.

The ultimate goal is not to reproduce HURDAT2 exactly, it is to obtain a physically meaningful storm center for transforming buoy observations into storm-relative coordinates. The suite therefore answers three separate questions:

Does the tracker follow the storm?
When does the tracker become unreliable?
How sensitive are storm-relative wave diagnostics to center uncertainty?

Framework

It is built around the idea that there is no single “true” center available at model resolution. Instead, several center estimates exist: Pressure minimum Vorticity maximum Composite cost-function minimum HURDAT2 best track

If all four agree, confidence should be high. If they diverge substantially, confidence should decrease. The suite therefore measures not only absolute error relative to HURDAT2, but also: internal consistency temporal smoothness physical plausibility sensitivity of storm-relative coordinates This provides a much stronger validation strategy than simply reporting back RMSE

Panel A - Track map and buoy locations

This is the first-order sanity check.

The panel overlays: HURDAT2 track Raw COAMPS center Smoothed center Confidence-weighted center Buoy locations

HURDAT2 represents operational analyses, aircraft fixes, satellite estimates and post-storm reanalysis. Meanwhile COAMPS-TC is evolving continuously at high temporal resolution. Therefore some differences are expected.

The most important observation is whether: the diagnosed center follows the same large-scale trajectory, the tracker remains on the storm, the smoothing procedure removes unrealistic jumps

Milton Very good agreement through most of the storm. Large divergence near the end. The tracker suddenly shifts westward relative to HURDAT2. The rest of the suite will tell us if this is tracker failure, physical ambiguity or model storm evolution differing from reality.

Idalia The cleanest case. The track is almost indistinguishable from HURDAT2 throughout most of the period. This is what you would expect from a well-defined compact vortex

Ian The tracker follows HURDAT2 extremely closely after the first few hours. The early discrepancy likely reflects spin up, center initialization or broad circulation structure. Once the vortex consolidates, the tracks are nearly identical.

Helene The most complicated case. Agreement is excellent until the final stage where a dramatic divergence develops. This is the largest displacement among all storms. Is the tracker wrong? or is the model storm simply no longer collocated with the observed storm? Later panels suggest the latter.

Panel B - Candidate center tracks

This panel compares: pressure minimum, vorticity maximum, composite tracker, and HURDAT2

If pressure and vorticity identify nearly the same location, the vortex structure is coherent. If they separate, the storm center becomes ambiguous.

Milton Strong agreement through most of the track. Near the end, candidate tracks begin to spread. That coincides exactly with increasing distance from HURDAT2, rising confidence penalty. This is strong evidence that uncertainty is real rather than numerical noise.

Idalia Pressure minimum and vorticity maximum nearly overlap. The vortex is compact and symmetric. Confidence should be high.

Ian Very good agreement. Small separations occur during structural evolution but overall coherence remains strong.

Helene The largest disagreement among candidate centers. Pressure and vorticity occasionally identify different locations. This indicates asymmetry, possible vortex elongation, or multiple competing circulation maxima.

Panel C - Distance to HURDAT2

This is a traditional error metric: $Δ r_{H} = ∣ x_{C O A M P S} - x_{H U R D A T 2} ∣$ The panel compares raw center, smoothed center and confidence-weighted center.

A large value can mean either track failure or the COAMPS storm differs from reality. These are fundamentally different conclusions and the rest of the suite exists to separate them.

Milton Mostly small errors with a late stage increase. This increase coincides with declining confidence. This combination makes the large distance believable rather than suspicious.

Idalia Mostly below 20-30 km.

Ian After initialization period errors are typically below 20km.

Helene Largest errors. Several periods exceed 50 km before the dramatic end-stage divergence. This tells us Helene deserves the most caution in later analysis.

Panel D - Confidence Score

The confidence metric combines multiple independent diagnostics: $C = w_{1} \frac{Δ r _{H}}{P _{95}} + w_{2} \frac{κ}{P _{95}} + w_{3} \frac{V _{c}}{P _{95}} + w_{4} \frac{( J _{2} - J _{1} ) ^{- 1}}{P _{95}}$ where track error, curvature, center velocity and cost-function ambiguity all contribute.

Physical meaning: A low C indicates smooth motion clear center strong candidate separation physically plausible trajectory A high C indicates competing centers abrupt jumps ambiguous vorticity structure poor temporal consistency

It evaluates track quality itself, not agreement with HURDAT2. A tracker can disagree with HURDAT2 and still have low ambiguity. Likewise it can agree with HURDAT2 while exhibiting obvious instability. Milton Early period The first half of the record is very stable Most values cluster around $C \approx 0.5 - 1.0$ This is very good The tracker is highly confident Middle period Confidence remains low despite occasional spikes Even though the storm evolves structurally, the tracker continues finding a consistent center Transition period Instead of isolated spikes you start seeing elevated background confidence, broader variability and clusters of moderate confidence This indicates persisted ambiguity Final period Confidence steadily increases. Many points exceed 2 or 3 This happens at the same time as panel c increasing upward, panel f developing huge radial offsets and candidate centers spreading apart. This is a good example of the confidence metric successfully identifying a deteriorating solution

Idalia Most of the storm The baseline remains steady around 0.8-1.3 for almost the entire record. This suggests a smooth center motion, strong candidate agreement and minimal ambiguity Spikes Occasional large spikes-> even a well behaved storm can have frames where pressure minimum shifts, vorticity maximum shifts or asymmetry temporarily increase. The confidence metric is sensitive enough to identify those events. End of record Confidence rises. Unlike the other storms, the baseline remains low and the increase is limited. The tracker remains trustworthy

Ian Beginning of record Elevated confidence values and many spikes This corresponds to the region where panel c shows larger HURDAT2 errors This may be because of initialization issues, broad circulation or center consolidation Main period Low confidence around d 0.5-1.0 The tracker appears smooth, stable and internally consistent. There are isolated occasional spikes, but the baseline never rises substantially Final period More spikes appear and moderate confidence events become more common, although the baseline remains fairly low The storm is becoming some what ambiguous

Helene Early period The confidence starts at 1-1.5 with numerous spikes. This suggests the tracker generally knows where the center is but alternative candidates appear frequently Short lived spikes can indicate transient asymmetry, competing local vorticity maxima or eyewall structure fluctuations rather than a complete tracker failure. Middle period The baseline drops with values often below 1. The center is moving slowly and candidate agreement is relatively strong. Later period Confidence increases systematically. Rather isolated spikes, elevated baseline, more frequent excursions and growing variability become evident. This could be a sign that the tracker is no longer struggling with isolated bad frames, but that the vortex itself is becoming harder to define. Final period The last several points jump toward 2-3 and above. This occurs simultaneously with track divergence, radial offset growth in panel f and large HURDAT distance in panel c. All three diagnostics deteriorate together, strongly suggesting the uncertainty is real.

Panel E - Vorticity snapshots

These images explain why confidence changes.

High confidence examples compact vortex, circular curvature and dominant vorticity maximum (the center is obvious) Ambiguous examples asymmetry, elongated core, or an offset between circulation center and vorticity maximum (several candidate centers become plausible) Very uncertain examples Fragmented circulation, open vorticity structures and strong deformation (no obvious center)

Easiest possible case: one dominant vorticity maximum, roughly symmetric circulation and an obvious eye. Hardest possible case: multiple maxima, elongated circulation, open vortex, deformation zones or competing features of similar strength. Milton Super confident cleanly defined eye, circular eyewall, strong annular vorticity ring and little deformation Pressure minimum, vorticity maximum and eye calmness all agree Ambiguous the eyewall is no longer perfectly circular, there is stronger wrapping asymmetry and a broader inner core structure The tracker can still find a center, but several nearby locations become nearly equivalent in cost -> $(J_{2} - J_{1})$ is small Very uncertain the vortex has disappeared. Instead yoy see a strong elongated vorticity ribbon, no dominant circular core and no obvious eye. This aligns with a large confidence penalty, exploding track error and large radial coordinate shifts.

Idalia Super confident the vortex is compact, isolated and nearly axisymmetric. Ambiguous ambiguity arises because of structural asymmetry -> the center is shifted toward one side, the eyewall is stronger on one flank, and the circulation is stretched. The vortex still exists but pressure and vorticity centers may not coincide perfectly. Very uncertain still have coherent rotation and an identifiable center region. The uncertainty appears to arise from weak gradients rather than a complete breakdown. This may explain why idalia maintains low confidence penalties overall.

Ian Super confident despite having much larger circulation, a broader eye, a wider vorticity ring and larger scale organization, the center still remains obvious The tracker is not dependent on compact storms, it works for larger systems too. Ambiguous very asymmetric circulation, the vortex still rotates around the mark location, but the strongest vorticity is displaced, the curvature center is displaced and eye geometry is distorted. Different diagnostics would choose different centers. Very uncertain the circulation still exists and the storm is still recognizable, but gradients weaken, the maxima broadens and the center region expands. There are now lots of of plausible centers (geometric rather than structural)

Helene Super confident near perfect vortex, strong circular symmetry and a dominant central maximum Ambiguous stretched vorticity structures, competing filaments and weak central organization Is there even a coherent center? Very uncertain the circulation exists, buy asymmetry is extreme, strong environmental interaction is evident and multiple spiral bands carry comparable vorticity The center becomes highly dependent on which metric is weighted most heavily

Confidence Metric appears physically meaningful, the classes correspond to different vortex states with a clear progression. The center uncertainty is not random, it occurs preferentially when the storm itself is transitioning, asymmetric, interacting with land or structurally evolving. This coincides with when storm relative wave interpretations become most vulnerable.

The figure demonstrates that the confidence metric is not merely detecting tracker noise, rather it is detecting genuine degradation in the underlying vortex structure.Overall, this panel provides the physical justification for why periods with elevated confidence penalties should be down weighted, flagged or excluded.

Panel F - Storm relative coordinate sensitivity

The upper panel measures $Δ r = r_{COAMPS} - r_{HURDAT2}$ where a positive value indicates that the buoy is farther from the storm center under COAMPS, and a negative value indicates the the buoy is closer to the storm center under COAMPS

The lower panel measures $Δ θ = θ_{COAMPS} - θ_{HURDAT2}$ (wrapped circularly) this measures changes in quadrant assignment, front/right vs rear/left designation, wind wave alignmnet calculations and azimuthal composites.

Across all storms, the radial coordinate is dramatically more stable than the azimuthal coordinate. if two centers are separated by a modest amount, the radius changes slowly but the azimuth can change rapidly. Especially when a buoy is close to the storm center. The large scatter in Δθ is not necessarily alarming.

Milton Δr for most of the record is <20 km, and the curves are nearly flat This means that the two centers would produce essentially identical radial classifications Late in the record it approaches -300 km. The transition is smooth rather than noisy. This suggests that there is a systematic center separation or the model storm diverges from HURDAT. This matches observations from panel C (explosion), confidence increase and candidate-track divergence. Δθ Before the divergence, the data is mostly clustered around 0° with occasional excursions. This means quadrant assignment is robust. After divergence, the scatter explodes and the data becomes unreliable.

Idalia Δr The curve is remarkably flat within +/- 20 km. Almost no meaningful radial sensitivity exists. It does begin to drift toward -200 km at the end of the record, but most buoy observations occur before then. Δθ The scatter is relatively large, but the central tendency remains around zero. This suggests that there is no systematic azimuth bias and only episodic quadrant sensitivity. This likely occurs when buoys lie near quadrant boundaries or when the radius is small.

Ian Δr During the initial period it begins at -150 km and approaches 0 with time. This corresponds to the initialization uncertainty identified in other panels. The tracker and HURDAT begin displaced before rapidly converging. For most of the record it is < 20 km. The tracker is reproducing the same storm relative radius. There are oscillations toward -30 -> -40 km, likely corresponding to eyewall changes, asymmetry or minor center shifts but remain small. Δθ Largest azimuth scatter of the four storms. There are multiple horizontal bands, suggesting many buoy points are repeatedly occupying similar geometric configurations. This is likely connect to fixed center offsets or repeated wraparound effects. The median still remains near zero, so there are no obvious systematic biases

Helene Δr For the first two thirds of the record it is <25 km, whcih is much better than peoiple would probably guess after examining panel C. Panel c measures center to center separation, while this panel measures buoy relative separation. A storm center can move 50 km while a distant buoy barely notices. In the final part of the record the radial offset smoothly increases toward -500 km. This aligns with the highest confidence penalties, the largest track errors and the most ambiguous vortex structures. (The suite is internally consistent) Δθ Most of the storm is clustered near 0° with moderate scatter. The final section develops distinct clusters. This is what you’d expect when two centers begin separating substantially. Small positional differences become amplified into large azimuthal differences.

Conclusion 1 For most of the buoy observations $∣Δ r ∣ < 25 km$ across all storms. This means radial wave analyses are highly robust to center uncertainty Conclusion 2 Azimuth is more sensitive than radius. This is expected geometrically. Radial composites and radius dependent trends are more trust worthy than exact quadrant boundaries. Conclusion 3 The periods of large uncertainty are concentrated. They are not distributed uniformly throughout the record. Instead, they occur during late Milton and Helene, and Ian initialization. This enables the flagging of only a small fraction of observations. Conclusion 4 The suite is internally self-consistent. The times where panel f deteriorates are the same times when panel c distances increase, panel d confidence rises and panel e vortex ambiguity increases. This agreement validates the framework an d provides strong evidence that the uncertainty estimates are physically meaningful rather than artifacts of the tracking algorithm.

Panel G -

This panel examines why the tracker chose the center that it did and which physical constraint was controlling the solution.

This panel plots each component of the center tracking algorithm : $J = w_{p} \overset{p}{^} - w_{ζ} \hat{ζ} + w_{u} \hat{U}_{eye} + w_{a} \hat{A} + w_{s} \hat{D}$ , while the black line plots the total cost

Across all storms the tracker is dominated by the pressure and vorticity terms.

Things to look for: Stable contributions -> the vortex is well defined Rapid switching -> potential ambiguity, multiple centers becoming competitive Smoothness becoming dominant -> the tracker may be relying on temporal continuity rather than physical structure

Milton Early storm pressure and vorticity remain stable, the curves are nearly flat. This implies a well defined vortex, little ambiguity and strong agreement between fields. This matches low confidence penalties, excellent coordinate consistency and clean vorticity snapshots Middle storm Gradual rise of the asymmetry contribution. The vortex becomes less idealized, yet the pressure and vorticity terms remain important. The tracker still has strong physical guidance. Final period The total cost begins to fluctuate dramatically. There are spikes, sign changes and increased variance. This corresponds to panel c divergence, panel d confidence growth, panel e uncertain vortex structure and panel f coordinate instability. Strong example of internal consistency

Idalia Entire storm Relatively stationary cost contributions and stable term influence End of record Asymmetry contribution starts to increase and total cost becomes less negative. This coincides with the small deterioration seen elsewhere in the suite. No component ever becomes dominant in an unstable way.

Ian Early storm Enormous variability (particularly vorticity contribution) rapid oscillations that align with elevated confidence, initialization uncertainty and larger HURDAT offsets. Main period Once the storm organizes, everything settles, the curves become smooth, and pressure and vorticity establish stable influence. This corresponds to Ian’s long interval of excellent tracking performance. Later period Asymmetry begins to grow, and total cost rises modestly This matches the moderate deterioration seen elsewhere

Helene Much more variability, more regime shifts and more abrupt transitions. First half Pressure and vorticity remain dominant, but they fluctuate more than in any other storm. The tracker is adjusting to structural changes. Around 2/29 The total cost jumps abruptly. Several contributions change regime simultaneously. This suggests more than tracker noise, but an actual change in storm structure Final period Multiple competing contributions, no single field dominates This corresponds to elevated confidence penalties, increasing center ambiguity and large coordinate sensitivity. The tracker is still functioning, but the underlying vortex becomes harder to define.

Conclusions The smoothness term never takes over. Contributions remain physically interpretable Panel progression G-> physical constraints become less coherent E-> vortex structure becomes less coherent D-> confidence decreases C-> track divergence increases F-> storm-relative coordinate sensitivity increases

Next steps

Save data: center_method = raw/smooth/confidence weighted C confidence_class delta_r_HURDAT delta_r_buoy = r_COAMPS-r_HURDAT delta_theta_buoy candidate_separation = J2-J1 storm_relative_quality_flag

Considerations use all points for exploratory plots use only low/moderate C points for primary results show that conclusions are unchanged when high C points are included flag or exclude late Helene, late Milton, and early Ian

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Notes - Jiang et al. 2022