Wyn Research Notes

Wyn Pauly

Physical Oceanography PhD · University of Hawaiʻi at Mānoa
Air–sea interaction · Tropical cyclones · Wave spectra · Directional organization

A living research notebook for methods, figure development, literature notes, and project documentation.

---

Research documentation

Methods

Technical documentation for analysis workflows, correction methods, validation logic, and physical interpretation.

Recent: Eulerian histories - wind only model

Project Log

Working interpretations, composite analyses, and emerging physical conclusions.

Recent: Wind-only equilibrium framework and residual wave-state variability

Literature

Paper walkthroughs, method notes, and conceptual background.

Recent: Effective Fetch and Duration of Tropical Cyclone Wind Fields Estimated from Simultaneous Wind and Wave Measurements: Surface Wave and Air–Sea Exchange Computation, Hwang and Fan, 2017

---

Updates

As of : Aug 13, 2026

Main development: moving from local forcing toward wave-following forcing history

This week I focused primarily on determining whether the departures from the instantaneous wind–wave relationship contain a recoverable signal of recent forcing history, and on developing a progressively more physical framework for representing that history.

The current analysis continues to use the validated wind-only GAM as a fixed baseline. Rather than replacing this model, I am treating its residuals as the quantity of interest and asking what additional information about wave state or forcing history explains departures from the local wind relationship.

The strongest result so far is that wave-following wind histories substantially improve the representation of MSS. A simple backward-propagated, peak-wave ray history increased explained deviance from approximately 0.78 to 0.88, corresponding to roughly 45% of the variance remaining after the instantaneous wind model. This is considerably larger than the improvements obtained from most purely local directional-state variables.

At the same time, the history timescale appears to matter. Comparing forcing windows showed that both the full previous 120 minutes and the older 60–120 min portion of the history outperform a history restricted to the most recent 0–60 min. This is particularly interesting for $f_p$, where the result suggests that the peak-scale wave field retains information from forcing substantially earlier than the instantaneous observation.

This moves the working interpretation away from simply “waves respond to an average recent wind” and toward a response that depends on the forcing encountered by propagating components of the spectrum.

Separating true memory from an alternative representation of present forcing

One important ambiguity remains: the historical wind estimate is still correlated with the contemporaneous wind. Therefore, improvement from the history term alone does not demonstrate that the GAM is responding uniquely to memory.

The next diagnostic is to isolate the historical component explicitly:

$$\Delta U_{\mathrm{ray}}=U_{\mathrm{ray},\mathrm{history}}-U_t$$

I will compare this against the equivalent Eulerian anomaly,

$$\Delta U_{\mathrm{Eulerian}}=U_{\mathrm{Eulerian},\mathrm{history}}-U_t$$

so that the question becomes:

Does MSS respond to recent changes in forcing, and is that history better represented by following the waves than by simply looking backward in time at the observation location?

This should provide a cleaner test of the Lagrangian interpretation than comparing the raw historical and instantaneous winds directly.

Frequency-dependent backward propagation

The current ray calculation follows a representative peak-wave component. The next major extension is to allow different portions of the spectrum to originate from different locations because of their different group velocities.

For deep-water waves,

$$c_g(f)=\frac{g}{4\pi f},$$

so a frequency-dependent history can be constructed without immediately moving to a substantially more complicated ray model.

The initial experiment will back-propagate several representative frequencies or frequency bands, for example:

• $f_p$
• $1.25f_p$
• $1.5f_p$
• $2f_p$
• representative high-frequency bands around approximately 0.25, 0.35, and 0.45 Hz

For each component I can sample the wind field along its trajectory over the same 0–60, 60–120, and 0–120 min intervals.

This provides an intermediate level of complexity between the current peak-ray estimate and full wave-ray calculations. I think it is the appropriate next step because it tests the central physical assumption that different spectral components have experienced different forcing histories before adding refraction, currents, or more sophisticated propagation physics.

Wind–wave alignment along the reconstructed history

A second major extension, prompted by our discussion of wind–wave alignment, is to retain the direction of the forcing encountered by each propagating wave component rather than using only wind speed.

For each point along a backward trajectory I can calculate

$$\Delta \theta (f,t)={\theta }_{\mathrm{wind}}-{\theta }_{\mathrm{wave}}$$

and the along-wave component of the wind,

$$U_{\parallel }=\mathbf{U}\cdot \hat{\mathbf{k}}=U\cos (\Delta \theta )$$

This may be particularly important for MSS and the high-frequency spectrum, where a large wind speed does not necessarily represent strong forcing of a wave component if the wind is strongly misaligned with its propagation direction.

The initial comparison will therefore retain several versions of the historical forcing:

1. scalar wind magnitude ($U$),
2. alignment angle ($\Delta \theta$),
3. signed along-wave forcing ($U_{\parallel }$),
4. positive-only along-wave forcing,
5. potentially an alignment-weighted integrated forcing.

The scalar wind history remains an important control so that any improvement attributable to directional information can be identified explicitly.

A later nondimensional version could compare $U_{\parallel }$ with the phase speed of the relevant component, but I would first establish whether the directional history itself contains additional information.

Evolving wave direction

The first frequency-dependent propagation experiment will probably assume that the observed propagation direction of each component remains fixed during backward reconstruction.

However, this raises a more interesting later experiment: allowing the propagation direction itself to evolve backward through the wind field.

Conceptually, this creates a hierarchy of increasingly physical history reconstructions:

frequency-dependent propagation → directional forcing along the ray → evolving wave direction → full ray dynamics/refraction.

This structure should make it possible to determine how much predictive information is gained at each level rather than moving immediately to a complicated ray calculation whose source of improvement would be difficult to diagnose.

Directional spreading as a state variable

I also tested whether directional spreading itself contains useful antecedent information.

The results so far suggest that directional spreading is more useful as a diagnostic of wave state/disequilibrium than as a universal predictor of memory.

Instantaneous HF directional spreading does not consistently improve the GAM, and antecedent HFDS produces relatively little robust improvement. Antecedent peak-frequency directional spreading appears more promising for $f_p$, but the improvement is not consistent across the other response variables.

This distinction is useful. The existing residual structures still suggest that directional organization changes systematically when the wave field is out of equilibrium with the instantaneous wind, but the spreading metric itself may not be the underlying forcing variable.

My current interpretation is therefore:

Directional organization is an observable signature of spectral memory, while the physical driver of that memory is more directly represented by the forcing history experienced by the propagating waves.

Directional reconstruction sensitivity

I also completed a more detailed comparison between the truncated-Fourier and MEM directional reconstructions to determine whether the directional results could be an artifact of reconstruction methodology.

Most response quantities are nearly insensitive to the reconstruction method. HFDS is the notable exception in absolute magnitude.

• Median HFDS increases from $4.36^{\circ }\to 5.46^{\circ }$ using MEM, an increase of approximately 20.7%.
• However, the relative variability is extremely well preserved: $r_{\mathrm{HFDS}}=0.97$.
• The wind-GAM HFDS residuals also remain strongly correlated between reconstruction methods: $r_{\mathrm{resid}}=0.91$.
• Spatial residual patterns are similarly stable, with approximately $r_{\mathrm{cell}}=0.95$.
• The MEM–Fourier differences increase modestly when the measured directional moments become weaker: ${\rho }_s=-0.24(R_1)$, ${\rho }_s=-0.22(R_2)$.

This suggests that reconstruction sensitivity behaves as expected when the directional information contained in the measured moments is weaker.

Most importantly, the relationship between HFDS residuals and the $H_s$/MSS residuals persists under both reconstruction methods. The reconstruction-dependent portion of HFDS itself has only a very weak relationship with those response residuals.

I therefore think the correct interpretation is that the absolute width of the reconstructed high-frequency distribution is method dependent, but the physically relevant relative HFDS variability and its residual structure are robust.

Relationship to the parametric trapping framework

I also spent some time thinking about the older parametric framework describing wave trapping within tropical cyclones.

There is an interesting conceptual connection between that work and the history framework I am developing. Both frameworks attempt to explain why the observed wave field cannot be understood from the instantaneous local wind alone.

The approaches differ substantially, however.

The parametric framework reduces the problem using bulk estimates of quantities such as wave momentum, storm translation, propagation, and drag. This makes the trapping concept physically intuitive and computationally inexpensive, but it necessarily introduces fairly strong assumptions about the wave field and forcing.

My current framework approaches the problem from the opposite direction: use the observed spectrum to reconstruct where different wave components are likely to have been and directly sample the modeled forcing along those histories.

A useful bridge between the two frameworks may therefore be a trapping parameter calculated from the reconstructed trajectories rather than assumed through bulk wave quantities.

For example, a diagnostic could quantify whether a given spectral component is effectively keeping pace with, remaining underneath, or escaping from the translating storm. That could eventually provide a physically interpretable scalar predictor alongside the direct history variables.

I would treat this as a later experiment, after establishing the frequency-dependent histories and alignment terms, rather than introducing another bulk parameter immediately.

Current hierarchy of experiments

The methodological progression is now approximately:

1. Establish frequency dependence

Back-propagate multiple spectral components using deep-water group velocities and determine whether their reconstructed forcing histories explain different response variables.

2. Add directional forcing

Calculate wind–wave alignment and along-wave forcing along each reconstructed trajectory.

3. Resolve the relevant memory timescale

Compare discrete lag windows and eventually construct lag-response surfaces rather than assuming a single integration period.

4. Test trapping diagnostics

Determine whether the relative propagation of waves and the translating storm provides an additional compact descriptor of the observed history.

5. Allow wave direction to evolve

Relax the fixed-direction trajectory assumption.

6. Move to more sophisticated ray calculations

Only after the above experiments establish which pieces of propagation physics actually contribute useful information.

The main response variables for these tests will initially be MSS, $R_{hf}$, $H_s$, and $f_p$. HFDS and peak-frequency directional spreading remain useful diagnostics but are probably secondary response variables for the forcing-history experiments.

Broader interpretation

The emerging picture is becoming increasingly consistent:

The instantaneous wind explains a substantial fraction of the wave-field variability, but observations with similar instantaneous forcing can occupy systematically different spectral states. Those departures contain information about earlier forcing, and that signal becomes substantially stronger when the forcing history is reconstructed from the perspective of the propagating waves.

The results also suggest that the wave field does not have a single memory timescale. Peak-scale properties appear sensitive to older forcing, while MSS and the high-frequency tail may be much more closely related to recent, directionally appropriate forcing.

This would naturally explain why directional organization is such a useful marker of disequilibrium: different portions of the spectrum can simultaneously reflect forcing accumulated over different trajectories and timescales.

Other research / infrastructure progress

I have also continued developing the virtual buoy / trajectory infrastructure alongside the analysis.

The trajectory system now provides a useful framework for generating and visualizing virtual buoy trajectories and associated forcing histories. This should eventually make the same trajectory logic being developed for the statistical analysis accessible interactively.

I have also been working through the architecture needed for live Spotter access and private data. Public Spotter data can be used to develop the viewer architecture first, while authenticated/private Spotter access can be added once the data-access layer is established.

The research site and associated data endpoints are now protected through Cloudflare Zero Trust, including the data required by the trajectory viewer. This required restructuring the cross-origin authentication behavior so that the browser can access protected research data while maintaining the login requirement.

The result is that the research tools can remain private without fundamentally changing the trajectory/data architecture.

Immediate next steps

My next analysis steps are:

• implement frequency-dependent backward propagation;
• retain wind direction and wave direction along each history;
• calculate scalar and along-wave forcing histories;
• compare 0–60, 60–120, and 0–120 min histories;
• repeat the GAM/residual tests using historical anomalies relative to the instantaneous wind;
• compare wave-following histories directly against equivalent Eulerian histories;
• identify whether MSS, $R_{hf}$, $H_s$, and $f_p$ favor different frequencies or memory timescales;
• only then consider evolving-direction or more sophisticated ray calculations.

The main question I am trying to answer next is no longer simply “does wind history matter?” The evidence increasingly suggests that it does.

The more useful question is:

Which parts of the spectrum remember which portions of the forcing history, and is that memory best explained by the trajectory and orientation of the waves relative to the evolving storm?

As of : Aug 4, 2026

Wind-only baseline framework

Over the past two weeks the focus shifted from data processing and coordinate validation to developing the first predictive framework for interpreting the observed wave state.

Rather than beginning immediately with forcing-history metrics, I first established an instantaneous “wind-only” baseline designed to quantify the component of wave variability that can be explained by the contemporaneous local environment alone. The intent is for this baseline to become the reference against which all subsequent history-dependent predictors are evaluated.

The framework currently models individual wave properties as smooth functions of the local atmospheric state using generalized additive models (GAMs), allowing nonlinear relationships without imposing a fixed functional form.

Model development

Considerable effort went into selecting a modeling framework that is both physically interpretable and robust across storms.

• Evaluated multiple spline basis functions (thin-plate, cubic regression, P-spline, shrinkage variants).
• Performed sensitivity analyses over spline complexity to balance flexibility against overfitting.
• Adopted leave-one-storm-out cross validation as the primary model selection criterion to ensure generalization across independent tropical cyclones.
• Selected parsimonious smoothness parameters using both predictive skill and stability of the fitted response functions.

The resulting framework provides a a reproducible baseline that avoids fitting specific behavior while remaining flexible enough to capture the nonlinear dependence of wave propriety on wind speed.

Response variable evaluation

A substantial portion of the work focused on selecting an appropriate representation for each response variable prior to fitting. Rather than applying transformations uniformly, each variable was evaluated using three complementary diagnostics: residual variance, distributional behavior (histograms and QQ plots), and leave-one-storm-out (LOOSO) predictive performance.

The resulting choices differed among variables.

• Significant wave height benefits from a logarithmic transformation, producing substantially more homogeneous residual variance while maintaining comparable predictive performance.
• Mean square slope is better represented in its raw form, which provides more consistent predictive skill across independent storms despite the skewed distribution.
• Peak frequency exhibits similar LOOSO performance in both representations, but the logarithmic transformation produces a noticeably more linear QQ relationship and was therefore adopted.
• Relative high-frequency energy ($R_{hf}$) performs best after logarithmic transformation, yielding both improved normality and consistently higher LOOSO explained deviance.
• High-frequency directional spread and peak directional spread remain poorly predicted by instantaneous forcing regardless of transformation. However, the logarithmic representation produces substantially more Gaussian residual distributions and improved QQ behavior, making it the preferred choice for subsequent analyses.
• Peak high-frequency alignment exhibits similar nonlinear residual structure under both representations, but the logarithmic transformation substantially reduces storm-to-storm variability in model skill. LOOSO explained deviance ranges from approximately 0.2 to -0.3 for the transformed variable compared to roughly 0.2 to -1.4 for the raw representation, indicating markedly improved robustness despite similar overall predictive performance.

Collectively, these diagnostics establish a consistent modeling framework while emphasizing that the preferred transformation depends on the statistical characteristics of each variable rather than on a single optimization criterion.

Residual analysis

With the baseline model established, attention shifted toward understanding the structure that remains unexplained.

Residuals were examined across all modeled wave properties rather than being treated simply as measures of prediction error. The objective was to determine which variables retain coherent, physically interpretable structure after accounting for concurrent wind forcing and storm-relative position.

The analysis reveals three distinct categories of behavior.

Bulk wave properties exhibit robust and repeatable residual structure. Significant wave height, mean square slope, and peak frequency all display systematic departures from the wind-only prediction that are consistent across independent storms. In particular, positive residuals in significant wave height and mean square slope occur preferentially during periods of weakening winds, while peak frequency tends to remain lower than predicted in similar conditions. These patterns suggest that the wave field retains memory of previous forcing, producing transient states that cannot be described solely by the local wind environment.

A second group of variables including relative high-frequency energy ($R_{hf}$) shows intermediate behavior. Although a substantial fraction of their variance is captured by the instantaneous model, coherent residual structure remains that may reflect changes in spectral organization and the adjustment of the high-frequency tail. The physical interpretation of these residuals is still under investigation, but they appear to respond on timescales distinct from the bulk wave properties.

In contrast, variables describing directional organization, including high-frequency directional spread, peak directional spread, and peak high-frequency alignment, exhibit consistently low predictive skill under the wind-only framework. Rather than displaying strong, repeatable residual regimes, these variables appear to contain relatively little variance that is directly attributable to instantaneous forcing. This suggests they may function primarily as descriptors of the evolving wave state itself rather than conventional response variables.

Taken together, these results suggest that instantaneous wind forcing explains the first-order evolution of many wave properties but does not uniquely determine the spectral state. The remaining variability is structured rather than random, motivating the explicit incorporation of forcing history into the analysis.

Current position

I now have:

1. a validated storm-relative reference framework with independently verified timing corrections;
2. a reproducible wind-only GAM framework with objectively selected response transformations, spline complexity, and leave-one-storm-out validation;
3. quantified baseline predictive skill for bulk and spectral wave properties spanning wave energy, spectral shape, and directional organization;
4. evidence that the principal residual structures are robust to reasonable response transformations, indicating that they reflect physical variability rather than statistical artifacts;
5. consistent residual correlation patterns across storms and storm quadrants, demonstrating that departures from the instantaneous wind-only prediction occur coherently across multiple wave properties;
6. an emerging interpretation in which the wind-only model defines an instantaneous equilibrium baseline, while the structured residuals describe departures associated with finite wave adjustment timescales and spectral evolution.

The project has therefore progressed from constructing an instantaneous predictive model to identifying coherent modes of residual variability that are reproducible across independent storms. Rather than representing unexplained statistical error, these residuals increasingly appear to describe physically meaningful departures from the local wind equilibrium and provide a quantitative target for the forcing-history analysis.

Planned next steps

The next phase will investigate whether the structured residual modes identified in the wind-only framework can be explained by the recent evolution of the local wind field.

Using the established instantaneous baseline, lagged and integrated forcing descriptors, including wind duration, cumulative forcing, directional rotation, persistence, and characteristic adjustment timescales, will first be evaluated individually against the residuals of each response variable. The goal is to determine which aspects of forcing history explain the coherent departures remaining after accounting for contemporaneous wind conditions.

A second objective is to investigate the role of directional wave organization within this framework. Because high-frequency directional spread, peak directional spread, and peak–high-frequency directional offset exhibit little direct dependence on instantaneous forcing, they will be examined as candidate descriptors of spectral state rather than conventional response variables. Their relationship to the residual modes of the bulk wave properties will help determine whether they encode the degree of spectral disequilibrium during wave-field adjustment.

Finally, the complete predictor set will be expressed in nondimensional storm-relative coordinates using candidate characteristic scales such as $U_{\max }$, $R_{\mathrm{mw}}$, and $R_{34}$. The resulting collapse in predictive performance and residual structure will determine which normalization best captures the common evolution of tropical cyclone wave fields across storms.

As of : July 22, 2026

Track correction and timing:

• Resolved the remaining HF-tail Doppler-correction inconsistencies and confirmed that the corrected spectra are now behaving as expected.

• Tested how the storm-center correction depends on the temporal relationship between the wind forcing and the observed HF wave orientation.

• Summary: The main comparison was between concurrent winds and a 60-minute lead. The geometric components of the optimization—rotation, ranking, displacement magnitude, and overall track shape—remained similar, while the directional matching term improved substantially with the lead. This showed that the experiment was primarily adjusting the temporal phasing between the wind and HF wave response rather than finding a fundamentally different storm track.

• Retained the 60-minute-lead correction, more precisely the t-60 to t-30 averaged HF-tail correction, as the final storm-relative coordinate framework.

• Validation:

  • The corrected track remains close to the best track and primarily introduces modest along-track phase adjustments.
  • Quadrant reassignments are concentrated near existing quadrant boundaries rather than reflecting wholesale changes in storm-relative geometry.
  • The directional improvement persists in a holdout experiment using data excluded from the fitting procedure.
  • Alternative energy-weighted and frequency-weighted alignment objectives produced nearly identical corrected tracks and did not improve the remaining quadrant variability.
  • The corrected quadrant composites recover coherent and repeatable TC-relative alignment structure across Ian, Idalia, Helene, and Milton.

• Quadrant analysis:

  • Generated wind-speed-binned spectral and alignment composites for all four storms using the corrected track and 30-minute-averaged concurrent winds.
  • The front-left, front-right, and back-right quadrants show the clearest improvements under the corrected framework. The back-left quadrant remains more variable, especially in the HF tail. I have avoided treating this as proof of a distinct quadrant-specific response timescale. Instead, it is now framed as residual structure that motivates analysis of forcing history.

• Normalization and forcing-history framework

  • Developed the broader normalization framework needed to compare observations across storms. The planned storm-relative dataset retains the native 15-minute atmospheric resolution and calculates, at each model time: $R_{\mathrm{mw}},R_{34},U_{\max },{\mathbf{V}}_{TC}$, along with normalized radius, azimuth, translation, and wind descriptors.
  • Decided to reconstruct the forcing at that fixed observed location backward through time and express it in the evolving normalized storm-relative coordinate system. This gives an Eulerian forcing history for every Spotter spectrum.
  • The immediate goal is to determine which parts of MSS, $R_{hf}$, HF directional spread, and alignment are explained by instantaneous conditions and which represent an additional history-dependent degree of freedom.

Current position

I now have:

1. a selected and independently validated storm-center correction
2. temporally consistent wind and Spotter averaging
3. corrected storm-relative coordinates and quadrant labels
4. coherent multi-storm validation figures
5. a defined normalization framework
6. a concrete plan for constructing Eulerian wind histories and fitting the instantaneous baseline before testing lag effects.

The project has shifted from deciding whether the coordinate and correction framework is trustworthy to using that framework to quantify wave-field memory and history dependence.

Planned next steps

The analysis will proceed in two stages. First, an instantaneous baseline will quantify how much of MSS, $R_{hf}$, HF directional spread, and alignment is explained by concurrent wind speed and storm-relative position. Second, lagged and integrated forcing descriptors will be tested against the baseline residuals to determine whether wind duration, rotation, persistence, cumulative forcing, or directional history explain additional variability.

Finally, the predictors and response variables will be nondimensionalized using candidate storm scales such as $U_{\max }$, $R_{\mathrm{mw}}$, and $R_{34}$. The degree of cross-storm collapse, predictive improvement, and residual structure will determine which normalization and history representation is most physically useful.

Tentative method approach : Representing Wind Forcing History for Wave Observations