Methods documentation: Buoy Corrections
Batch Diagnostics
Figure 1 - Invalid-bin fraction versus forcing/correction variables
This figure asks a simple question: under what physical conditions does the Doppler correction cease to be well behaved?
This figure relates the fraction of invalid bins within each spectrum to several environmental and wave-state variables, including buoy drift speed, maximum projected drift normalized by group velocity, peak period, and significant wave height.
The strongest relationship appears between invalid-bin fraction and which serves as the most direct measure of Doppler correction strength. As the projected buoy drift approaches the group velocity of the waves, the fraction of invalid bins increases rapidly. This behavior is expected because strong drift causes increasingly aggressive frequency remapping, pushing the transformation toward its mathematical limits.
Relationships with peak period and significant wave height are noticeably weaker. Although larger waves often correspond to stronger winds and larger buoy drift velocities, neither parameter directly controls the correction itself. The results therefore suggest that the relevant control parameter is not overall storm intensity but rather the relative motion between the buoy and the wave field. This finding supports the theoretical framework of the Doppler correction and identifies as the key diagnostic variable governing correction reliability.
Figure 2 - Minimum Jacobian Diagnostics (remake)
The left panel shows the minimum Jacobian value encountered in each spectrum, while the right panel summarizes the overall distribution across all storms. The Jacobian provides a direct measure of local compression or expansion of the frequency axis induced by buoy drift.
Most spectra exhibit minimum Jacobian values between approximately 0.4 and 1.0. Values near unity indicate weak Doppler influence, while smaller positive values indicate increasing compression of intrinsic frequency space into observed frequency space. Importantly, the vast majority of spectra remain positive, confirming that the transformation remains physically invertible over most of the dataset.
The distribution plot shows that negative Jacobian values are rare outliers rather than a dominant feature of the correction. This is a major improvement over earlier diagnostic versions, where extreme negative values suggested numerical instability. After removing those artifacts, the Jacobian distribution now reflects physically plausible behavior. The remaining negative cases represent the small subset of spectra for which buoy drift approaches or exceeds the group velocity of the shortest waves, producing genuine breakdowns of the frequency mapping.
Figure 3 - MSS response versus correction strength
This figure quantifies how the Doppler correction alters mean square slope (MSS) as a function of correction strength. The left panel uses maximum relative frequency shift, while the right panel uses maximum projected drift normalized by group velocity,
Both panels reveal a remarkably coherent trend: stronger Doppler corrections systematically increase MSS. This behavior is physically consistent with the expected effect of buoy drift. Forward buoy motion shifts high-frequency waves toward lower observed frequencies, artificially reducing spectral energy at the frequencies that contribute most strongly to MSS. Correcting for this drift restores that missing high-frequency variance, leading to larger slope estimates.
The trend is particularly striking because it appears across multiple storms and spans a broad range of environmental conditions. MSS increases are typically modest for weak corrections but can exceed factors of 1.5–2 when projected drift approaches wave group velocity. The smooth, monotonic nature of this relationship suggests that the correction is recovering a physically meaningful signal rather than introducing random numerical noise. This figure therefore provides some of the strongest evidence that the Doppler correction has a systematic and interpretable impact on high-frequency wave statistics.
Figure 4 - First Jacobian-failure frequency by failure mode
This figure shows the frequency at which the Doppler correction first becomes invalid for each spectrum, separated by the specific failure mechanism. Failures are categorized as (1) Jacobian sign reversals , (2) non-positive mapped spectral energy, (3) non-finite interpolation or mapping values, and (4) other miscellaneous numerical failures. The shaded region highlights the 0.2–0.5 Hz band that contributes most strongly to MSS and high-frequency tail diagnostics.
The most important result is that the majority of invalid frequencies occur through the “nonfinite mapped term” pathway rather than through Jacobian sign reversal. True Jacobian failures are relatively uncommon and tend to occur only at the highest frequencies in the analysis band. This indicates that most invalid bins arise from numerical limitations of the frequency remapping and interpolation procedure rather than from a fundamental breakdown of the Doppler transformation itself. The correction therefore remains mathematically well-posed for most spectra, even when substantial portions of the high-frequency band are ultimately discarded.
The distribution of failure frequencies also reveals storm-to-storm differences. Idalia failures tend to occur at lower frequencies, whereas Helene and Milton frequently remain valid until frequencies approaching 0.4–0.5 Hz. This suggests that differences in buoy drift speed, wave age, and wave direction relative to buoy motion strongly influence the frequency range over which the correction remains usable.
Combined interpretation
The Doppler correction substantially modifies high-frequency wave statistics and MSS in energetic tropical cyclone conditions, but the correction remains mathematically valid for the vast majority of spectra. Most invalid bins arise from numerical issues associated with aggressive frequency remapping rather than from widespread Jacobian sign reversal. The primary controlling parameter is the ratio of projected buoy drift speed to wave group velocity, , which governs both correction strength and the likelihood of invalid high-frequency bins.