Wind velocity estimates from wave observing platforms

ABSTRACT

Near-surface ocean wind measurements are important for weather forecasting, determining surface transports, and estimating air-sea interactions; however, in-situ wind observations are often limited. Previous studies indicate that the equilibrium range of the surface gravity wave spectrum can be used to estimate surface wind velocity. This approach is tested using spectral wave measurements from the Coastal Data Information Program (CDIP) buoy array off Monterey, California. Quality controlled wind vectors inferred from wave spectra are statistically compared to measurements from a nearby National Data Buoy Center (NDBC) buoy, demonstrating strong agreement with the control observations with root mean square errors for speed and direction of 1.8 m/s for all wind speeds and 13.2° for wind speeds greater than 7 m/s. We expand this estimation method to account for biofouling, which causes high-frequency damping of the wave spectrum, and the effects of form factor, which impact the platform’s dynamic response to high-frequency waves. The method produces wind-proxy measurements solely from wave spectra and wave-based drag parametrizations, making it useful for operational integration. This work demonstrates the ability to make robust wind velocity estimates using wave data from multiple sources, increasing the coverage of wind information over the coastal and open ocean.

KEYWORDS:

• Surface gravity waves
• wind waves
• wind
• buoy observations
• in-situ oceanic observations

1. Introduction

In coastal environments, meteorological and ocean conditions often vary on small scales difficult to resolve with spatially dispersed in-situ measurements, temporally infrequent remote-sensing observations, or models. For example, the National Data Buoy Center (NDBC) hosts an array of meteorological buoys actively used to inform the public, authorities, and emergency services of marine weather conditions; however, operation and maintenance costs limit buoy coverage to broad regional scales. In pursuit of additional wind information, recent work has shown that spectral wave measurements can be used to estimate wind speed and direction (Shimura et al. 2022; Thomson et al. 2013; Voermans et al. 2020), supplementing the limited coverage of oceanic wind measurements. Several of these methods rely on direct wind estimations to derive drag coefficients or wave age parameters, which is impractical for operational applications. Relative to meteorological stations, wave buoys are often cheaper and simpler to deploy (Herbers et al. 2012). Recent technological developments have led to an increase of reliable wave measurements from autonomous platforms (Amador, Merrifield, and Terrill 2023), expanding the coverage of wave observations and subsequently the possibility of wind information.

Short surface gravity waves are closely coupled to the surface wind field as the majority of momentum is transferred from the wind to surface waves at high frequencies (Phillips 1985). The evolution of ocean surface waves is governed by the conservation of wave action, which is defined as

(1) $\frac{\mathit{dN}}{\mathit{dt}}=S_{\mathit{tot}}=S_{\mathit{in}}+S_{\mathit{diss}}+S_{\mathit{nl}}$(1)

where $\mathit{N}=\mathit{E}/\mathit{f}$ is the wave-action density, $\mathit{E}$ is the wave energy, and $\mathit{f}$ is the frequency. The source term, $S_{\mathit{tot}}$, is the sum of wind input, $S_{\mathit{in}}$, dissipation due to breaking, $S_{\mathit{diss}}$, and transfer via nonlinear wave-wave interactions, $S_{\mathit{nl}}$. Phillips (1985) suggests that for a high-frequency portion of the wave energy spectrum, these source and sink terms balance, such that an equilibrium range can be found when $S_{\mathit{tot}}=0$. From this, Phillips (1985) derived the following analytic expression for the energy spectrum in this range

(2) $\mathit{E}\left(\mathit{f}\right)=E_0{f}^{-4}\mathrm{where}{E}_0=\frac{4\mathrm{g\beta I}\left(p\right)u_{\ast }}{{\left(2\mathit{\pi }\right)}^3}$(2)

where $u_{\ast }$ is the wind friction velocity, $\mathit{g}$ is gravitational acceleration, $\mathit{\beta }$ is an empirical constant, and $\mathit{I}(\mathit{p})$ is the directional spreading function. The directional spreading function varies between 1.9 and 3.1 (Juszko, Marsden, and Waddell 1995), where larger values indicate a narrower directional spread. Equation (2)(2) $\mathit{E}\left(\mathit{f}\right)=E_0{f}^{-4}\mathrm{where}{E}_0=\frac{4\mathrm{g\beta I}\left(p\right)u_{\ast }}{{\left(2\mathit{\pi }\right)}^3}$(2) indicates that in the equilibrium range, the wave spectrum will have an expected slope of $f^{-4}$ and energy that is proportional to the wind friction velocity. The wind dependence of wave spectra in the equilibrium range is illustrated using wind speed and wave spectra observations at Monterey Bay (Figure 1), where wave energy at high frequencies scales with the contemporaneous wind speed with a $f^{-4}$ dependence.

Figure 1. Wave spectra observed at CDIP station 185. The color corresponds to the wind speed observed at the time of measurement. A slope of $f^{-4}$ is denoted by the dashed line.

Previous work has demonstrated the ability to use this relationship to infer wind speed from wave spectra observed from a range of wave measurement platforms (e.g Jiang 2022; Thomson et al. 2013; Voermans et al. 2020). Thomson et al. (2013) used Ocean Station Papa wave data from a 0.9-meter diameter Datawell Directional Waverider (DWR MKIII) buoy and three Surface Wave Instrument Floats with Tracking (SWIFTs) and compared wind friction velocity estimates to direct anemometer-measured wind velocities. Voermans et al. (2020) estimated wind velocities over a three-year period at more than 100 NDBC sites equipped with 3-meter diameter wind-wave buoys to verify the integrity of the method over a range of geographic conditions and sea states. Voermans et al. (2020) also applies the method to a smaller, 40-centimeter diameter Spotter wave buoy (Raghukumar et al. 2019), finding that the wave measurement platform size has an impact on the wind speed estimation. This finding motivates further work to test the reliability of this approach for different platforms, especially if estimated wind speeds from multiple platforms will be combined when using them for climatologies or forecast assimilation. In this study, we confirm that the application of Equation (2)(2) $\mathit{E}\left(\mathit{f}\right)=E_0{f}^{-4}\mathrm{where}{E}_0=\frac{4\mathrm{g\beta I}\left(p\right)u_{\ast }}{{\left(2\mathit{\pi }\right)}^3}$(2) works well for Datawell Directional Waverider buoys used by the Coastal Data Information Program (CDIP) with reliable high-frequency wave measurements; however, we identify discrepancies when dealing with measurements from unique platforms such as Wave Gliders and biofouled buoys.

In the open ocean, the frequency range of interest for sea/swell surface gravity waves is approximately $0.05<\mathit{f}<0.5$ Hz. The shortest timescale that a floating body can respond to is the heave response frequency, $f_{\mathit{hr}}$, given by Hudspeth (2006),

(3) $f_{\mathit{hr}}=\frac{1}{2\mathit{\pi }}\sqrt{\frac{F_B\mathit{g}}{\mathit{ML}}}$(3)

where $F_B={\rho }_{\mathit{sw}}\mathit{V}$ is the buoyancy, $\mathit{V}$ is displacement volume, ${\rho }_{\mathit{sw}}$ is the density of seawater, $\mathit{M}$ is mass, $\mathit{L}$ is the vertical length of the body, and $\mathit{g}$ is gravitational acceleration. The observing platform’s heave response varies as a function of size ($\mathit{L}$), mass ($\mathit{M}$), and geometry. In this paper we describe methods for inferring wind velocities from spectral wave measurements from platforms with varying form factors.

The paper is organized as follows. In Section 2 we describe the method used to estimate wind velocities from surface wave spectra. The measurements are described in Section 3. In Section 4.1 we validate the wind estimates from Coastal Data Information Program (CDIP) operated Datawell Directional Waverider buoys and compare the estimates to direct wind measurements. We then discuss the effect of form factor on the wind estimate and propose additional quality control metrics. Section 4.2 applies the method to NDBC wind-wave buoy data. It is shown that for Wave Glider produced wave spectra, the method can be applied with corrections to heave attenuation that occurs in high-frequency ranges due to the platforms geometry and inertia (Section 4.3). We then show that for Waverider buoys with severe biofouling, the high-frequency response to surface waves is damped, and the observed equilibrium range is narrowed (Section 4.5). In both cases, due to the methods adaptive capabilities and the introduction of equilibrium quality control criteria, wind velocities can be estimated from the wave observations, demonstrating the versatility of the method across multiple wave measurement platforms.

2. Methods

2.1. Wind estimation from wave spectra

2.1.1. Equilibrium range

The equilibrium range is determined based on the observed wave spectrum at each 30-minute (one hour for NDBC wave data) time interval. Equation (2)(2) $\mathit{E}\left(\mathit{f}\right)=E_0{f}^{-4}\mathrm{where}{E}_0=\frac{4\mathrm{g\beta I}\left(p\right)u_{\ast }}{{\left(2\mathit{\pi }\right)}^3}$(2) implies that the wave spectrum $\mathit{E}(\mathit{f})$ normalized by $f^4$ will have a predicted slope of 0 in the equilibrium range and be proportional to wind friction velocity.

To determine the equilibrium frequency range, a fitting method is applied to the spectral data to determine the slope of $\mathit{E}(\mathit{f})f^4$ for different frequency combinations beyond a low frequency cutoff. We expect that $\mathit{E}(\mathit{f})f^4$ will have a slope of 0 within the equilibrium range. Since wave spectra are dominated by remotely forced swell at low frequencies, a cutoff is set at $2f_p$ (where $f_p$ is the frequency of the peak energy of the spectrum) since this is the region where spectral energy follows $\mathit{E}(\mathit{f})\propto u_{\ast }{f}^{-4}$ (Toba 1973). The linear regression slope is computed for each contiguous segment of the normalized energy spectrum [$\mathit{E}(\mathit{f})f^4$] with band widths ranging from 0.14 to 0.34 Hz (corresponding to 15 to 35 frequency indices), producing slope values corresponding to every possible combination of starting frequencies and frequency band widths within the $2f_p-f_{\mathit{max}}$ range, where $f_{\mathit{max}}$ is the highest frequency resolved by the energy spectrum for each platform. At each time interval, an average value of $\mathit{E}(\mathit{f})f^4$ is computed across the fitted frequency band with a regression slope closest to 0.

An example of the equilibrium range selection method is shown in Figure 2. From here, wind friction velocity at each time can easily be determined by rearranging Equation (2)(2) $\mathit{E}\left(\mathit{f}\right)=E_0{f}^{-4}\mathrm{where}{E}_0=\frac{4\mathrm{g\beta I}\left(p\right)u_{\ast }}{{\left(2\mathit{\pi }\right)}^3}$(2) as follows

Figure 2. (a) Wave energy spectra from a CDIP buoy (blue) and the predicted equilibrium slope (dashed black). (b) An example of the equilibrium frequency range fitting method showing $\mathit{E}(\mathit{f})f^4$ (blue), upper and lower limits of the selected frequency range using the fitting method (vertical black) and the average $\mathit{E}(\mathit{f})f^4$ value over that range (red), used to compute $u_{\ast }$.

(4) $u_{\ast }=\frac{\overline{E(f)f^4}{(2\mathit{\pi })}^3}{4\mathit{\beta I}(\mathit{p})\mathit{g}}$(4)

where $\overline{E(f)f^4}$ indicates the average of $\mathit{E}(\mathit{f})f^4$ over the selected equilibrium range frequencies. Following Thomson et al. (2013), $\mathit{I}(\mathit{p})$ = 2.5 is used, corresponding to waves in the equilibrium range that are aligned with the wind. The constant $\mathit{\beta }$ is estimated empirically from wind speed and wave age observations, and is set as $\mathit{\beta }=0.012$ following Juszko, Marsden, and Waddell (1995).

2.1.2. Wind speed and direction

The surface wind stress $\mathit{\tau }$ can be expressed as

(5) $\mathit{\tau }=\mathit{\rho }{u}_{\ast }^2=\mathit{\rho }{C}_D{U}_{10}^2$(5)

where $U_{10}$ is the 10-m wind speed, $\mathit{\rho }$ is the density of air, and $C_D$ is a non-dimensional drag coefficient (e.g. Large and Pond (1981); Smith (1980)). The drag parameterizations used in this study follow the COARE 3.5 algorithm described by Edson et al. (2013), which considers wave age and wave slope dependence as well as wind speed dependence on the Charnock coefficient $\mathit{\alpha }$ and has a sole input of $T_p$, the peak wave period, which is a standard wave platform output. Wind direction is determined by averaging the wave direction values across the same frequency range used for determining wind speeds.

2.1.3. Logarithmic equilibrium slope criteria

Once the equilibrium frequency range is determined, a logarithmic equilibrium slope of the energy spectrum is computed within that range. Typical values of this slope are near −4, which is the expected slope in the equilibrium range. However, in cases of high or low wind speeds, heave attenuation, or damping, there may be no high-frequency region of the spectrum that sufficiently presents the expected −4 slope that can be used to provide reliable proxy wind information (shown in Figures 1 and 4). Hence, equilibrium slope criteria can be set to filter the data, serving as an indicator of the degree to which a given energy spectrum represents the expected $f^{-4}$ equilibrium slope. Proxy values corresponding to equilibrium slopes outside of the range of −3.6 to −4.4 are removed hereafter.

3. Data

This study examines three unique observational wave platforms with varying form factors and heave response frequencies. In Section 3.1 we introduce the CDIP wave buoy data, and in Section 3.2 we present the NDBC buoy wind and wave data used to validate the CDIP wind proxy measurements. In Section 3.3 we introduce a new experimental platform, a Wave Glider, and explain the limitations of its high-frequency wave observations. Finally, in Section 3.4, we examine an instance of extreme biofouling to a CDIP buoy, which alters its heave response frequency.

3.1. CDIP wave observations

CDIP manages an array of over 80 wave buoy stations around the United States, with directional wave data dating back to the 1990s. The buoys provide reliable, long-term wave measurements for use in planning, designing, and operating coastal projects. As a result, quality controlled CDIP measurements are released to the public in near-real time for use by coastal engineers and planners, scientists, mariners, and marine enthusiasts.

This study uses spectral wave data collected at CDIP station 185, (NDBC station 46114) using a 0.9-meter diameter Datawell Directional Waverider MKIII buoy (Datawell 2019) located roughly 50 kilometers west-northwest of Monterey, California (Figure 3). The buoy was moored in 1,463-meter water depth at ${36.700}^{\circ }$N ${122.343}^{\circ }$W. The mooring is connected to the Waverider with a rubber cord and a weighted line below to create a “false bottom” allowing for the buoy to accurately follow surface wave motions.

Figure 3. Data stations used in this study, located offshore of Monterey Bay. The buoys are separated by approximately 11 km.

Accelerometer-measured pitch, roll, and heave displacement values are recorded at 1.28 Hz for approximately 26 minute time intervals (Datawell 2019) and reported every half hour. Spectra are calculated onboard using eight 200 second long windows with no overlap, resulting in 0.01 Hz frequency resolution and 16 degrees of freedom. The wave energy and direction are calculated in 64 frequency bands over a range of 0.0250–0.5800 Hz. The spectra are transmitted via Iridium satellite to CDIP at Scripps Institution of Oceanography, where they are made publicly available in near-real time. A year of data are used in this study, collected between January 1 2021 and December 31 2021.

3.2. NDBC wind and wave observations

Wind and wave data were collected at NDBC station 46042 using an NDBC standard 3-meter diameter discus buoy with a seal cage (NDBC 2009). The buoy was moored in 1,693-meter water depth at ${36.785}^{\circ }$N ${122.396}^{\circ }$W. The distance between the CDIP and NDBC buoys was approximately 11 kilometers, shown in Figure 3.

NDBC reported wave data come from accelerometer measurements recorded each hour (NDBC 2009). A Fast Fourier Transform is applied to vertical acceleration data on board to convert into the frequency domain. Response amplitude operator processing is applied to account for hull and electronic noise, acceleration spectra is converted to displacement spectra, and wave parameters are released publicly online. The resulting wave energy and directional data is resolved into 47 bands with a frequency range of 0.0200–0.4850 Hz. The temporal resolution of the NDBC data (every hour) is less than that of the corresponding CDIP data (every half hour), and the frequency vector of the CDIP data extends into higher ranges compared to NDBC (0.5800 Hz versus 0.4850 Hz). The frequency resolution of the NDBC spectral data is nonlinear (frequency steps increase slightly beyond 0.35 Hz), so the energy density and wave direction data are linearly interpolated to match the frequency increments of the CDIP wave data for consistency.

Wind estimates from both the CDIP and NDBC wave observations are compared to the NDBC anemometer wind measurements. Wind measurements are converted to the standard height of 10 m following

(6) $U_{10}/U_a=(10/z_a{)}^{0.11}$(6)

from Hsu, Meindl, and Gilhousen (1994) where $z_a$ is the anemometer height of 3.8 m and $U_a$ is the measured wind speed at that height. The average wind speed and direction is the vector average of the observations over 8-minute intervals. The wind speed and direction values are averaged to match the temporal resolutions of the wave spectral data, which is 30-minute and one-hour intervals for CDIP and NDBC data, respectively.

Compared to a Datawell Directional Waverider CDIP buoy, an NDBC 3-meter Discus buoy has an increased size (L) and mass (M), and the high-frequency response of the buoy is reduced following Equation (3)(3) $f_{\mathit{hr}}=\frac{1}{2\mathit{\pi }}\sqrt{\frac{F_B\mathit{g}}{\mathit{ML}}}$(3) . This is demonstrated in Figure 4, where the spectral tail of the NDBC data rolls off beyond 0.35 Hz and does not assume the expected $f^{-4}$ equilibrium slope beyond this threshold.

Figure 4. Comparison of averaged CDIP (blue), NDBC (green), wave glider (red), and wave glider with heave attenuation corrections (black) wave spectra.

3.3. Wave glider wind and wave observations

Observations of winds and waves were gathered from May 22 2020 through July 10 2020 off the coast of Southern California by an SV3 Wave Glider (Hine et al. 2009, Figure 5(a)). The vehicle traveled offshore past San Nicolas Island (Figure 5(b)), and circled CDIP station 067 (integrated into the NDBC as station 46219) during June 12-18 2020 while maintaining a distance ranging from 1 to 8 km from the buoy (yellow dot in Figure 5(b)). The Wave Glider was instrumented with the Coastal Observing Research and Development Center (CORDC)-fabricated environmental sensing payload (Amador, Merrifield, and Terrill 2023).

Figure 5. (a) Liquid Robotics SV3 Wave Glider and (b) the vehicle’s path (black) as it moves offshore, circles CDIP station 067 (yellow dot) and continues past San Nicolas Island.

Bulk wind measurements were collected 1.2 m above the sea surface using a sonic anemometer (Vaisala WXT520) installed on a vertical mast on the Wave Glider float. Wind samples were collected at a sampling frequency of 1 Hz and averaged over 10-minute intervals, followed by 4 minutes of onboard processing and telemetry, resulting in measurements approximately every 14 minutes. Observed wind speeds were motion-compensated and adjusted to a standard height of 10 meters using Equation (6)(6) $U_{10}/U_a=(10/z_a{)}^{0.11}$(6) .

Wave measurements were obtained using a GPS-directional wave sensor mounted on the float. The wave data underwent processing procedures following the methodology outlined in Amador, Merrifield, and Terrill (2023), including the conversion of velocity time series (512-s bursts) into frequency spectra and the subsequent correction for Doppler shifting. The resulting spectra and cospectra were resolved into 247 bands with a frequency range of 0.0391–1 Hz, corresponding to a period of approximately 25.6–1.0 s. The sea surface elevation spectra, $\mathit{E}(\mathit{f})$, were then estimated from the Doppler-corrected vertical velocity auto-spectra via linear wave theory. Wave directional moments and the associated mean wave direction were computed following the work of Herbers et al. (2012), with minor modifications.

To facilitate comparisons across measurement platforms, the Wave Glider sea surface elevation spectra and mean wave direction values were linearly interpolated to match the frequency resolution of a Datawell Directional Waverider, similarly to the NDBC data preparation. Wave data were averaged to match the temporal resolution of the wind data ($\approx 14$ minutes). Vehicle speed and direction are not adequately defined when turning, which prevents Doppler shift corrections; thus, the associated data were discarded. This criterion limited the number of valid spectra to about 97% of the total bursts. As noted by Amador, Merrifield, and Terrill (2023), Wave Glider observations of wave spectra exhibit moderate attenuation at high frequencies. Thus, concurrent $\mathit{E}(\mathit{f})$ data from CDIP 067 are used to account for the vehicle’s muted heave response (section 3.3.1).

3.3.1. Heave attenuation correction in wave glider observations

Comparisons between the CDIP buoy and Wave Glider spectral data indicate heave attenuation in the high-frequency range $\mathit{f}>0.25$ Hz (Figure 6), likely a result of the float’s relative size and inertia (Amador, Merrifield, and Terrill 2023). Because the attenuation occurs in the equilibrium range used for the wind proxy estimate, adjustments are needed.

Figure 6. Comparison of (a) the wave glider (black) and CDIP (blue) wave spectra and (b) the average of $E_{\mathit{CDIP}}/E_{\mathit{WG}}$ for all realizations during the time period when the two platforms were within 20 kilometers of each other. Underestimations of the wave glider produced wave spectra must be accounted for before applying the wind-proxy method since it relies heavily on accurate energy measurements within the equilibrium range.

The Wave Glider energy density data are scaled using the ratio between the average CDIP buoy and Wave Glider energy density values for each frequency band. This method may not be effective when there is no nearby control wave buoy data that can be used for scaling.

3.4. Biofouled wave buoy observations

Wave spectral data come from two Datawell Directional Waverider MKIII 0.9-meter diameter buoys located at Ocean Station Papa, which has been an ocean reference station in the North Pacific (roughly ${50}^{\circ }$N ${145}^{\circ }$W) since the 1940s (Freeland 2007). Current Waverider measurements are incorporated into the CDIP as station 166 and the NDBC as station 46246.

In October 2012, a Waverider buoy was deployed at Ocean Station Papa and in January 2015, the buoy was scheduled for a swap due to suspected biofoul accumulation over its two year deployment. When the replacement buoy was deployed, foul weather caused a delay in the recovery of the existing mooring, resulting in two collocated CDIP buoys collecting data simultaneously. A 42 hour window of overlapping data from the old (biofouled) and new (clean) buoys, deployed within 20 km of each other, is used in this study. The wave data are determined using the same methods as described for CDIP station 185.

In cases of extreme biofouling the size (L) and mass (M) of the buoy are effectively increased, which attenuates the high-frequency spectral response and affects estimates of wave direction (Figure 7). The clean buoy has a response frequency of approximately $f_{\mathit{hr}}=1$ Hz following Equation (3)(3) $f_{\mathit{hr}}=\frac{1}{2\mathit{\pi }}\sqrt{\frac{F_B\mathit{g}}{\mathit{ML}}}$(3) and using a mass of $M_0=225$ kg, displaced volume which is half that of the buoy $V_0=\frac{1}{2}[(\frac{4}{3})\mathit{\pi }(\frac{D}{2}{)}^3]=0.425$ $m^3$, and a vertical length $L_0=0.45$ m for a 0.9-meter Cunifer hull buoy. The estimated volume of the biofouled buoy $V_{\mathit{bf}}=1.425$ $m^3$ and estimated vertical length $L_{\mathit{bf}}=1.5$ m yield a response frequency for the biofouled buoy of $f_{\mathit{hr},\mathit{bf}}=0.35$ Hz (Thomson et al. 2015).

Figure 7. Spectral data (a) and mean wave direction (b) for the clean (blue) and biofouled (red) buoys throughout their deployment. Clear deviations beyond 0.35 Hz for wave energy and direction indicate damping of high-frequency waves as a result of decreased heave responsiveness for the biofouled buoy.

The energy density values for the biofouled buoy drop-off significantly and deviate from the $f^{-4}$ equilibrium slope beginning around 0.35 Hz, which is typically within the equilibrium range and is consistent with its corresponding response frequency $f_{\mathit{hr},\mathit{bf}}$. Because the wind proxy method relies heavily on these high-frequency data, we explore quality control criteria to account for high-frequency rolloff.

4. Results

4.1. CDIP

A portion of the full time series for direct- and proxy-measured wind speed (Figure 8(a)), shows the largest discrepancies when direct-measured $U_{10}$ is weak (see also Figures 9(a,c)). The proxy wind measurements exhibit negative bias (Figure 9(b)) and directional errors of up to ${60}^{\circ }$ for low direct-measured wind speeds (Figure 9(d)). 2.6% of wave spectra are removed by defining equilibrium slope criteria requiring values between −3.6 and −4.4 for a sufficient fit.

Figure 8. Time series comparisons for direct-measured (red), NDBC proxy-measured (green), and CDIP proxy-measured (blue) wind speed (a) and direction (b) for a portion of the dataset used in this study. Highest errors in the proxy-measured speed and direction values occur when wind speeds are low (< 3 m/s).

Figure 9. (a) Measured wind speeds from NDBC 46042 plotted against calculated wind speeds using wave observations from CDIP 185. The least squares regression line between measured and estimated winds is shown in blue. The RMSE between direct-measured and proxy-measured values for all wind speeds is 1.8 m/s. (b) Bias (black) and RMSE (red) values over all wind speeds. (c) Measured direction values plotted against calculated wind direction values (colored by direct-measured wind speeds). (d) Direction RMSE and mean error for all wind speeds.

For wind speeds of relatively low magnitude ($<3$ m/s) the proxy-measured relative error becomes large, with significant $U_{10}$ underestimations. The RMSE between direct-measured and proxy-measured values is 1.8 m/s for all wind speeds and 1.7 m/s for wind speeds 3–12 m/s. For wind speeds greater than 12 m/s, the percent relative error is 15%. Table 1 summarizes the CDIP wave buoy proxy-results, which are comparable to the results from Voermans et al. (2020).

Table 1. Comparison of wind speed results.

Platform	RMSE all $U_{10}$	RMSE $U_{10}$ 3–12 m/s	Percent error $U_{10}>12$ m/s
CDIP 185	1.8 m/s	1.7 m/s	15%
NDBC 46042	2.0 m/s	2.0 m/s	19%
Wave Glider	1.9 m/s	1.8 m/s	19%

Similarly to wind speed, the RMSE between direct-measured and proxy-measured wind directions is high at low wind speeds (Figure 8(b)), increasing from ${13.2}^{\circ }$ for winds speeds >7 m/s, to ${56.2}^{\circ }$ for wind speeds <7 m/s. The RMSE is ${14.2}^{\circ }$ for wind speeds >11 m/s and ${43.4}^{\circ }$ for all wind speeds (Table 2).

Table 2. Comparison of wind direction results.

Platform	RMSE $U_{10}>7$ m/s	RMSE $U_{10}<7$ m/s	RMSE $U_{10}>11$ m/s	RMSE all $U_{10}$
CDIP 185	${13.2}^{\circ }$	${56.2}^{\circ }$	${14.2}^{\circ }$	${43.4}^{\circ }$
NDBC 46042	${48.1}^{\circ }$	${68.9}^{\circ }$	${54.0}^{\circ }$	${60.1}^{\circ }$
Wave Glider	${34.2}^{\circ }$	${41.4}^{\circ }$	${47.9}^{\circ }$	${38.8}^{\circ }$

4.2. NDBC

Based on the equilibrium slope criteria, 22.7% of the NDBC spectra were not considered. The RMSE between direct-measured and proxy-measured values for all wind speeds is 2.0 m/s for all wind speeds and 2.0 m/s for wind speeds 3–12 m/s. For wind speeds larger than 12 m/s, the percent relative error is 19%. The RMSE are slightly higher for NDBC than CDIP wave data. Figure 8 shows that NDBC proxy-measured wind speeds are frequently underestimated compared to the direct-measured and CDIP proxy-measured values.

Directional errors are also increased compared to CDIP wave directional data, with a RMSE of ${48.1}^{\circ }$ for wind speeds >7 m/s, ${68.9}^{\circ }$ for wind speeds <7 m/s, ${54.0}^{\circ }$ for wind speeds >11 m/s, and ${60.1}^{\circ }$ for all wind speeds.

4.3. Wave glider

The RMSE between direct-measured and proxy-measured values for all wind speeds is 1.9 m/s, for wind speeds 3–12 m/s the RMSE is 1.8 m/s, for wind speeds >12 m/s the percent relative error is 19% (Table 1). Based on the equilibrium slope criteria, 29.5% of proxy measurements are eliminated. The RMSE between direct-measured and Wave Glider proxy-measured wind direction is ${34.2}^{\circ }$ for wind speeds >7 m/s, ${41.4}^{\circ }$ for wind speeds <7 m/s, ${47.9}^{\circ }$ for wind speeds >11 m/s, and ${38.8}^{\circ }$ for all wind speeds (Table 2).

4.4. Comparison of results

Table 1 compares wind speed results between measurement platforms, indicating largest errors for all proxy wind measurements using NDBC buoy observations, likely due to the platforms increased size and decreased high-frequency response. Table 2 summarizes similar results for wind proxy direction values between platforms, with larger errors for NDBC and Wave Glider proxy results compared to CDIP, validating the reliability of high-frequency CDIP wave directional data.

The high-frequency edge of the equilibrium range selected from the regression slope fitting can demonstrate how the method is able to avoid unreliable high-frequency spectral data and is denoted as $f_{\mathit{edge}}$ hereafter. By allowing the selected frequency bands to vary in width and $f_{\mathit{edge}}$, the method adapts to a range of spectra between various observational platforms and conditions. The method tends to select a lower $f_{\mathit{edge}}$ for the Wave Glider and NDBC buoy because these platforms have lower roll-off frequencies compared to the CDIP buoy (Figure 10). Similarly, the NDBC buoy is more likely to have a narrower equilibrium band width selected because of limited high-frequency data and decreased heave responsiveness.

Figure 10. Probability density of each (a) $f_{\mathit{edge}}$ and (b) frequency band width being selected for NDBC (green) CDIP (blue) and wave glider (black) observational platforms. Variations indicate the versatility of the equilibrium selection method through its adaptation to different platform spectra.

4.5. Biofouling

The average logarithmic equilibrium slope values for the biofouled and clean buoy data are −6.83 and −4.00, respectively, demonstrating that the slope of the selected frequency range can be an indicator of spectral fit to the expected $f^{-4}$ equilibrium. By setting equilibrium slope criteria to filter the quality of equilibrium fit, 56% of biofouled buoy proxy-measurements can be eliminated.

For spectra corresponding wind proxy values that do or don’t meet the equilibrium criteria, the RMSE between the clean and biofouled buoy wind speed values is 1.3 and 8.5 m/s, respectively. The comprehensive results shown in Tables 3 and 4 demonstrate the method’s ability to be selective in the wave spectra that can be used to make adequate wind estimates.

Table 3. Clean vs biofouled buoy wind speed results.

Logarithmic equilibrium slope (r)	RMSE all $U_{10}$	RMSE $U_{10}$ 3–12 m/s	Percent error $U_{10}>12$ m/s
$-4.4<\mathit{r}<-3.6$	1.3 m/s	1.2 m/s	13%
$\mathit{r}<-4.4$ or $\mathit{r}>-3.6$	8.5 m/s	7.7 m/s	81%

Table 4. Clean vs biofouled buoy wind direction results.

Logarithmic equilibrium slope (r)	RMSE $U_{10}>7$ m/s	RMSE $U_{10}<7$ m/s	RMSE $U_{10}>11$ m/s	RMSE all $U_{10}$
$-4.4<\mathit{r}<-3.6$	${12.8}^{\circ }$	${64.8}^{\circ }$	${9.9}^{\circ }$	${42.6}^{\circ }$
$\mathit{r}<-4.4$ or $\mathit{r}>-3.6$	${12.1}^{\circ }$	${94.5}^{\circ }$	${11.1}^{\circ }$	${49.8}^{\circ }$

Figure 11(a,b) shows the probability of each $f_{\mathit{edge}}$ and band width being selected by the fitting method across the clean buoy and biofouled buoy datasets. For wave spectra corresponding to a logarithmic equilibrium slope that don’t meet the criteria, we see that the selected frequency band decreases in width and increases in $f_{\mathit{edge}}$. Instances where the method is forced to select a high $f_{\mathit{edge}}$ value where it is impossible to achieve equilibrium because of the biofouled buoys inability to accurately measure high-frequency energy produce significantly underestimated wind speeds, shown in Figure 11(c). The clean buoy has a wider spread for both metrics, as it is not effected by high-frequency damping.

Figure 11. The probability density of each possible (a) $f_{\mathit{edge}}$ and (b) frequency band width being selected by the method for the clean buoy (blue) and the biofouled buoy where the logarithmic equilibrium slope meets the criteria (red) or doesn’t (green). (c) Clean buoy proxy wind measurements plotted against biofouled buoy proxy wind measurements colored by the logarithmic equilibrium slope of the data within the fitted frequency band selected. Values near the expected equilibrium slope (−4) tend to have accurate proxy measurements and wind speeds are underestimated for biofouled buoy proxy measurements where the equilibrium slope does not meet the fit criteria.

5. Discussion

Our method requires a range of the wave spectrum to be in equilibrium, which may not exist at very high or very low winds. We find that our method becomes unreliable at high (>12 m/s) and low wind speeds (<3 m/s) (Figure 9). For moderate wind speeds, ocean surface momentum exchange is dominated by form drag, and a majority of momentum is transferred to waves (eg. Edson et al. (2007)). For very low wind speeds, momentum exchange between the ocean and atmosphere is dominated by viscous drag (Kudryavtsev and Makin 2001), and a majority of momentum is transferred to surface currents. For these low wind speeds, it’s likely that equilibrium assumptions do not apply, and thus wind stress for mild winds cannot be determined solely from wave spectral data. Similarly, for more energetic seas, equilibrium assumptions may not reflect the evolution of the equilibrium range for larger wave heights or breaking waves. The presence of swell can also modify this relationship by affecting either the equilibrium range (e.g. Thomson et al. (2013), García-Nava et al. (2012)) or the drag coefficient (e.g. Donelan and Dobson (2009), Potter (2015)), which can affect the accuracy of the wind estimation more drastically in low wind conditions or in mature, swell-dominated seas. Our study suggests that the range of wind speeds that can be accurately estimated from wave spectral data are between 3–12 m/s. Our results expand on the methods described by Thomson et al. (2013) and Voermans et al. (2020) by using a drag coefficient independent of direct wind measurements and by determining an equilibrium slope criteria.

This method assumes constant air-sea coefficients ($\mathit{I}(\mathit{p})$ and $\mathit{\beta }$), though accounting for variations may increase the methods accuracy. $\mathit{\beta }$ has been shown to vary with mean wave steepness and wave age (e.g. Voermans et al. 2020), and consideration for these variations may reduce error in the estimated wind speeds. All wind speeds were adjusted to a standard height of 10 meters following Equation (6)(6) $U_{10}/U_a=(10/z_a{)}^{0.11}$(6) . A logarithmic wind profile above the sea surface is best described by a Charnock parameter $\mathit{\alpha }$ that is estimated from a surface roughness term $z_0$ and also varies with wave steepness and wave age, indicating that multivariable parameterizations of $\mathit{\alpha }$ may improve wind speed estimates. However, these parameters cannot be derived without direct measured wind speeds.

Newer Waverider buoys are typically coated with antifouling paint, drastically reducing biological growth and subsequently decreasing instances of damping of high-frequency heave responsiveness. However, the equilibrium slope criteria is useful when using this method to produce long-term climatologies from historical data where biofouling is more common.

6. Concluding remarks

This study validates the wind estimate method for CDIP wave buoys and applies it to datasets where considerations must be taken into account for platforms with variable high-frequency heave responses to ocean waves. It is shown that the accuracy of the estimated wind speeds using wave spectral data alone is sufficient for several applications and can be used when no direct wind measurements are available. The proxy method yields estimated wind speeds with an overall RMSE value of 1.8 m/s, 2.0 m/s, and 1.9 m/s for CDIP wave buoy, NDBC wind-wave buoy, and Wave Glider measured wave spectra, respectively. The overall RMSE between direct measured and proxy measured wind direction for the CDIP wave buoy, NDBC wind-wave buoy, and Wave Glider is ${43}^{\circ }$, ${61}^{\circ }$, and ${39}^{\circ }$, respectively. This demonstrates the improved versatility of this method by allowing the selected frequency band width to vary in order to account for platform form factor and compromised high-frequency data, increasing the number of collocated wind and wave data observations available to scientists, mariners, and marine enthusiasts.

Heave responsiveness varies between measurement platform primarily due to size, geometry, and inertia. By introducing equilibrium slope criteria, the proxy method can be applied in instances when the observational platforms high-frequency heave responsiveness is compromised, as in the case of extreme biofouling or limited high-frequency data. The equilibrium slope is reflective of the quality of the equilibrium range selection and provides adequate filtering of the data that does not demonstrate an equilibrium range. While performing slightly worse than the other assets, Wave Glider wind proxy results show that that the Wave Glider can provide reasonable solutions, with comparable accuracy to other more traditional platforms. Also, this indicates that the Wave Glider heave attenuation technique (Section 3.3.1) can be a reliable approach for correcting wave spectra in the equilibrium range. In general, high-frequency energy spectra are more accurate for smaller buoys, shown by the reduced wind vector error from NDBC to CDIP buoy wind proxy results. This demonstrates the advantage of using CDIP measured wave spectra for determining wind proxy values because of the relative compactness of CDIP buoys.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the U.S. Army Corps of Engineers.