Buoy Corrections

The goal of the correction is to estimate the intrinsic surface-wave spectrum from spectra measured by drifting buoys.

A drifting buoy does not observe waves in a stationary reference frame. Because the buoy moves with currents, stokes drift and wind slip, the observed frequencies are doppler shifted relative to the intrinsic frequencies of the waves. This effect is most important at high frequencies, where many wave diagnostics (particularly MSS) are highly sensitive to small frequency errors.

Correction procedure

The correction procedure consists of:

1. Estimating buoy drift relative to the water
2. Computing the Doppler shift associated with that drift
3. Transforming the observed spectrum into an intrinsic spectrum
4. Recomputing spectral diagnostics from the corrected spectrum

Step 1 : Estimate buoy drift velocity

The buoy velocity is decomposed into several components: ${\mathbf{U}}_{\mathrm{buoy}}={\mathbf{U}}_{\mathrm{current}}+{\mathbf{U}}_{\mathrm{Stokes}}+{\mathbf{U}}_{\mathrm{slip}}$ where $U_{current}$ is the eulerian surface current $U_{Stokes}$ is the wave-induced stokes drift $U_{slip}$ is direct wind forcing on the buoy

Wind slip is estimated using comparisons between spotter and microswift drift rates. The assumed relationship is approximately $U_{\mathrm{slip}}=\alpha U_{10}$ with $\alpha \approx 0.012$ for the excess spotter drift relative to microswift.

Step 2 : Project drift onto wave direction

Only the component of buoy motion along the direction of wave propagation contributes to the Doppler shift.

The projected drift is $U_{\mathrm{proj}}={\mathbf{U}}_{\mathrm{buoy}}\cdot \hat{\mathbf{k}}$ or equivalently $U_{\mathrm{proj}}=U_{\mathrm{buoy}}\cos ({\theta }_{\mathrm{wave}}-{\theta }_{\mathrm{buoy}})$ where ${\theta }_{\mathrm{wave}}$ is the wave propagation direction ${\theta }_{\mathrm{buoy}}$ is the buoy drift direction

The notebook uses an MSS-weighted wave direction because MSS is dominated by the short waves most strongly affected by Doppler shifting.

Step 3 : Doppler frequency correction

For a moving observer -> ${\omega }_{\mathrm{obs}}={\omega }_{\mathrm{int}}+\mathbf{k}\cdot \mathbf{U}$ or ${\omega }_{\mathrm{obs}}={\omega }_{\mathrm{int}}+k{U}_{\mathrm{proj}}$ where ${\omega }_{\mathrm{obs}}$ is observed angular frequency, ${\omega }_{\mathrm{int}}$ is intrinsic angular frequency, $k$ is wavenumber.

The correction seeks to recover ${\omega }_{\mathrm{int}}$ from ${\omega }_{\mathrm{obs}}$ using the estimated projected drift velocity.

Step 4 : Use the dispersion relation

The Doppler term depends on wavenumber, which depends on frequency through the linear gravity-wave dispersion relation. For deep water : ${\omega }^2=gk$ which implies $k=\frac{{\omega }^2}{g}$ Substituting into the Doppler equation gives ${\omega }_{\mathrm{obs}}={\omega }_{\mathrm{int}}+\frac{{\omega }_{\mathrm{int}}^2}{g}{U}_{\mathrm{proj}}$ which must be solved for the intrinsic frequency corresponding to each observed frequency bin.

Step 5 : Transform spectral density

Because the frequency coordinate changes, spectral energy density must be transformed using a Jacobian.

Energy conservation requires $E_{\mathrm{int}}({\omega }_{\mathrm{int}})d{\omega }_{\mathrm{int}}=E_{\mathrm{obs}}({\omega }_{\mathrm{obs}})d{\omega }_{\mathrm{obs}}$ which yields $E_{\mathrm{int}}=E_{\mathrm{obs}}\left|\frac{d{\omega }_{\mathrm{obs}}}{d{\omega }_{\mathrm{int}}}\right|$ The Jacobian is therefore $J=\frac{d{\omega }_{\mathrm{obs}}}{d{\omega }_{\mathrm{int}}}$ and the corrected energy becomes $E_{\mathrm{corr}}=E_{\mathrm{raw}}J$ This is where the negative-energy issue arises if the Jacobian becomes negative.

Step 6 : Interpolate onto a regular frequency grid

After correction, frequencies no longer lie on the original Spotter frequency bins.

The corrected spectrum must therefore be interpolated back onto a standard frequency grid: $E_{\mathrm{corr}}(f)\to E_{\mathrm{corr}}(f_{\mathrm{grid}})$ for subsequent analysis.

This interpolation step is often where pathological spectra appear if: the frequency mapping becomes non-monotonic, the Jacobian approaches zero, corrected energies become negative.

Step 7 : Recompute wave diagnostics

…

Invalid Bins

The implicit assumption behind the Doppler correction is : There exists a unique intrinsic wave spectrum that can be recovered from the observed spectrum through a smooth coordinate transformation. $f_{\mathrm{obs}}=f_{\mathrm{obs}}(f_{\mathrm{int}})$ must remain 1:1 $J=\frac{d{f}_{\mathrm{obs}}}{d{f}_{\mathrm{int}}}>0$ everywhere The moment J crosses zero, that assumption breaks.

The typical regime in which this correction is applied occupies $\frac{U_{\mathrm{proj}}}{c_g}\sim 0.05-0.2$, where the correction is a perturbation and the transformation is gentle (the Jacobian stays near 1).

In these cases, the buoy is moving at a substantial fraction of the speed at which short-wave energy propagates. The observer is no longer sampling the wave field, it is dynamically embedded within it.

Invalid bins during the strongest parts of observations provide information about the physical regime: strong currents, strong stokes drift, strong buoy drift and very short waves such that $U_{\mathrm{proj}}\sim c_g$ for a substantial part of the 0.2-0.5 Hz band. This is an extreme observational environment.

The strongest corrections occur when $\frac{U_{\mathrm{proj}}}{c_g}\to 1$ which coincides with when MSS is largest, the tail is most energetic and wave-supported stress is likely the most important. The frequencies we are trying to resolve are exactly where the observational geometry becomes least trustworthy. This is a fundamental limitation of trying to infer intrinsic short-wave properties from a rapidly drifting buoy in a tropical cyclone.

Corrected high-frequency diagnostics are used only where the Doppler transformation remains monotonic and sufficiently well conditioned. Where the correction fails within 0.2–0.5 Hz, raw diagnostics are retained and the observation is flagged as outside the valid correction regime.

Reported products: ${\mathrm{raw}}_{0.2-0.5}$ ${\mathrm{corr}}_{0.2-f_{\max ,valid}}$ $\mathrm{quality}\mathrm{flag}$

Summary

Doppler correction was attempted following Davis et al. However, during periods of strongest tropical cyclone forcing, the projected buoy drift velocity became comparable to the group velocity of short gravity waves (U_{\rm proj}/c_g \approx 1). Under these conditions the frequency transformation became poorly conditioned or non-monotonic, resulting in negative Jacobians and invalid spectral bins at high frequency. Consequently, corrected diagnostics were only computed where the frequency mapping remained monotonic and sufficiently conditioned.