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.”

Storm-Relative Coordinate System and Track Correction

Directional wave spectra were analyzed in a storm-relative reference frame using collocated Spotter buoy observations and COAMPS-TC atmospheric fields. Storm-relative coordinates were computed from the vector connecting the corrected tropical cyclone center to each buoy location and projected onto a coordinate system defined by the storm translation direction. The along-track coordinate was defined positive in the direction of storm motion, while the cross-track coordinate was defined positive to the right of the translation vector. Storm-relative angle and radius were then used to derive quadrant classifications when required.

The storm center positions were obtained using the buoy-constrained track correction described previously. Briefly, high-frequency tail alignment was used to estimate small storm-center displacements relative to the modeled track, while temporal regularization constrained unrealistic jumps and preserved smooth storm motion. Validation experiments showed that applying the correction using winds averaged over the preceding 60–30 minute interval produced the most spatially coherent storm-relative wave organization across multiple storms. This lead-time correction was therefore adopted for all subsequent analyses.

Because observations located near quadrant boundaries remain sensitive to small uncertainties in storm position and heading, analyses primarily utilize the continuous storm-relative coordinates (radius, azimuth, along-track, and cross-track position), with quadrant composites used only as a secondary qualitative diagnostic.

Spectral Doppler Correction

Observed directional spectra were corrected for Doppler shifting associated with the combined platform drift and surface current following the processing framework developed by Davis et al. Relative platform velocity was estimated as the sum of the Spotter Stokes drift and a small wind-driven slip velocity proportional to the local high-frequency-tail wind. Observed spectra were then transformed from encounter frequency to intrinsic frequency through a frequency remapping procedure that conserves spectral variance via the appropriate Jacobian transformation.

Validation against the reference processing demonstrated conservation of integrated spectral variance while allowing physically consistent shifts in spectral peak frequency. As expected, the correction redistributes energy across frequency without altering total variance, providing a more physically meaningful representation of wave age, peak frequency, and high-frequency spectral properties. All subsequent analyses were therefore performed using the Doppler-corrected intrinsic-frequency spectra.

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.