Track Correction

Multi-buoy spectrally constrained track correction

Using the HF tail as a local wind field proxy -> The highest reliable frequencies are the part of the wave spectrum most tightly coupled to recent/local wind forcing, so their direction and organization should evolve coherently with the modeled wind field.

Approach

${\theta }_{HF}(b,t)$ -> HF mean wave direction $\dot{{\theta }_{HF}}(b,t)$ -> temporal rotation

Then fit one center correction per time: ${\delta }_t=[\Delta x_t,\Delta y_t]$ by minimizing $\mathcal{J}={\sum }_t{\sum }_{b\in B_t}\left[w_{\theta }{J}_{\theta ,b,t}+w_{\dot{\theta }}{J}_{\dot{\theta },b,t}\right]+{\lambda }_1{\sum }_t||{\delta }_t-{\delta }_{t-1}|{|}^2+{\lambda }_2{\sum }_t||{\delta }_t-2{\delta }_{t-1}+{\delta }_{t-2}|{|}^2$ where : $J_{\theta ,b,t}=1-\cos \left({\theta }_{HF,b,t}-{\theta }_{U,model,b,t}({\delta }_t)\right)$ and : $J_{\dot{\theta },b,t}=1-\cos \left({\dot{\theta }}_{HF,b,t}-{\dot{\theta }}_{U,model,b,t}({\delta }_t)\right)$ Use signed directional tendency over angle vs angle matching $\dot{\theta }=\frac{\mathrm{unwrap}({\theta }_t)-\mathrm{unwrap}({\theta }_{t-1})}{\Delta t}$ so that : $J_{\dot{\theta },b,t}={\left({\dot{\theta }}_{HF,b,t}-{\dot{\theta }}_{U,b,t}^{model}({\delta }_t)\right)}^2$ This captures whether the direction is steady, rotating clockwise, or rotating counterclockwise at the right time.

Then the objective becomes: $\mathcal{J}={\sum }_t{\sum }_{b\in B_t}\left[w_{\theta }\left(1-\cos ({\theta }_{HF}-{\theta }_U)\right)+w_{\dot{\theta }}{\left({\dot{\theta }}_{HF}-{\dot{\theta }}_U\right)}^2\right]+{\lambda }_1||\Delta {\delta }_t|{|}^2+{\lambda }_2||{\Delta }^2{\delta }_t|{|}^2$ First term : does the corrected model wind point the same way as the observed HF tail? Second term : does the corrected model wind rotate in the same way as the HF tail? ${\lambda }_1$ : prevents jumps ${\lambda }_2$ : prevents wiggles

The direction term will be dominant, with the rotation term as a secondary constraint.

In summation : The correction is constrained only by the high-frequency directional tail, which is treated as the most locally wind-coupled part of the spectrum. The optimized track correction is the smooth displacement that makes the modeled wind direction and modeled wind-direction rotation agree best with the observed HF-tail direction and rotation across all buoys.”

Results

Through trouble shooting, the displacement-prior term was removed due to its suppression of real corrections. With this term removed the RMSE between model wind direction and buoy HF wave direction and rotation improved by: Ian -> 2.08 deg, 100% of buoys Idalia -> 4.66 deg, 100% pf buoys Helene -> 2.12 deg, 78.6% of buoys Milton -> 4.09 deg, 91.3% of buoys

Data Quality Control and Storm-Relative Coordinate Sensitivity

Directional wave spectra were analyzed in a storm-relative reference frame using collocated Spotter buoy observations and hurricane track information. Storm-relative coordinates were computed from the vector connecting the interpolated storm center to each buoy location and projected onto a coordinate system defined by the instantaneous storm translation direction. The along-track coordinate was defined as positive in the direction of storm motion, while the cross-track coordinate was defined positive to the right of the storm-motion vector. Quadrant assignments were then derived algebraically from the signs of the along-track and cross-track coordinates, yielding front-right (FR), front-left (FL), back-right (BR), and back-left (BL) classifications.

During validation of the processed Idalia dataset, the stored quadrant labels were compared against quadrant assignments recomputed directly from the storm-relative coordinate geometry. The underlying storm-relative coordinate values were found to be internally consistent with the buoy and storm positions; however, the stored quadrant labels exhibited systematic disagreement with the algebraic classification. Only 68.8% of observations matched the stored labels, with the dominant discrepancies corresponding to left-right reversals. In particular, many observations labeled as back-left were geometrically consistent with back-right positions, while a substantial fraction of front-right observations were geometrically consistent with front-left locations. Because the coordinate values themselves remained consistent, the discrepancy was attributed to the stored categorical labels rather than the underlying storm-relative coordinate calculations.

To evaluate the robustness of storm-relative classifications, a sensitivity analysis was performed using multiple estimates of storm translation direction. Quadrant assignments were recomputed using heading vectors calculated over a range of temporal windows. Additionally, the angular distance between each observation and the nearest quadrant boundary was calculated. Observations were classified as heading-sensitive if their quadrant assignment changed across reasonable storm-motion windows, and as boundary-ambiguous if their storm-relative angle lay within a prescribed margin of a quadrant boundary.

Of the 480 valid processed Idalia observations, 227 retained the same quadrant assignment across all tested heading windows and were therefore classified as stable. Twenty-five observations exhibited heading-window sensitivity, indicating dependence on the specific choice of storm-motion estimate. The remaining 253 observations were located sufficiently close to a quadrant boundary that small changes in storm center position or heading estimate could alter their classification. Consequently, more than half of the available observations were found to possess ambiguous quadrant membership despite being associated with well-defined storm-relative coordinates.

These results indicate that hard quadrant classifications should be interpreted cautiously when applied to buoy observations collected near storm-track boundaries. While storm-relative coordinates themselves remain physically meaningful, categorical quadrant labels may not represent robust dynamical regimes when observations cluster near dividing lines between sectors. Accordingly, storm-relative angle, along-track position, and cross-track position were treated as higher-confidence diagnostics than quadrant membership alone. Quadrant composites were retained for qualitative interpretation but were supplemented by analyses based on continuous storm-relative variables and by sensitivity tests using only observations with stable quadrant assignments.

Spectral Doppler Correction and Frequency Remapping

Observed buoy spectra were corrected for Doppler shifting associated with relative motion between the wave field and the observing platform. The correction was implemented through a frequency remapping procedure in which observed frequencies were transformed to intrinsic frequencies using the estimated relative propagation velocity. A Jacobian transformation was applied to preserve spectral variance during the coordinate transformation.

To verify consistency with previous processing approaches, corrected spectra were compared against independently processed spectra generated using the original workflow. Matched-spectrum comparisons demonstrated near-perfect conservation of zeroth spectral moment, with corrected and remapped spectra differing negligibly in integrated variance. For representative observations, both processing methods produced identical values of spectral variance to within numerical precision, indicating that the correction procedure conserved total wave energy.

Although integrated variance remained unchanged, remapped spectra frequently exhibited modest shifts in peak frequency. Analysis of matched observations showed that these shifts originated primarily during the Doppler-remapping stage rather than during subsequent interpolation to a common frequency grid. Across all matched Idalia observations, the median ratio between remapped and reference peak frequency was 1.023, with the central 80% of cases lying between 0.970 and 1.071. Interpolation to a common frequency grid introduced only minimal additional peak-frequency changes.

The distinction between variance conservation and peak-frequency preservation is important. The Doppler correction redistributes energy across frequency while preserving the total spectral variance. Consequently, peak location and local spectral density may change even when integrated energy remains constant. For analyses emphasizing physically meaningful wave scales, such as peak frequency, wave age, spectral tail behavior, and high-frequency energy content, the intrinsic-frequency representation provides a more physically consistent description of the wave field than spectra expressed solely on the observed frequency coordinate.

How the high frequency tail is directionally organized relative to the peak

Bulk sea-state metrics such as peak frequency and mean square slope constrain the first-order structure of the wave field, but do not uniquely determine the organization of the high-frequency tail. Residual variability in tail alignment, directional spreading, and energy partitioning may therefore contribute to variability in wave-supported stress and air-sea momentum transfer that is not captured by bulk metrics alone.