An empirical model for wind-generated ocean noise

An empirical model for wind-generated underwater noise is presented that was developed using an extensive dataset of acoustic field recordings and a global wind model. These data encompass more than one hundred years of recording-time and capture high wind events, and were collected both on shallow continental shelves and in open ocean deep-water settings. The model aims to explicitly separate noise generated by wind-related sources from noise produced by anthropogenic sources. Two key wind-related sound-generating mechanisms considered are: surface wave and turbulence interactions, and bubble and bubble cloud oscillations. The model for wind-generated noise shows small frequency dependence (5 dB/decade) at low frequencies (10–100 Hz), and larger frequency dependence ( 15 dB/decade) at higher frequencies (400 Hz–20 kHz). The relationship between noise level and wind speed is linear for low wind speeds (<3.3 m/s) and increases to a higher power law (two or three) at higher wind speeds, suggesting a transition between surface wave/turbulence and bubble source mechanisms. At the highest wind speeds (>15 m/s), noise levels begin to decrease at high frequencies (>10 kHz), likely due to interaction between bubbles and screening of noise radiation in the presence of high-density bubble clouds. VC 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1121/10.0005430 (Received 22 February 2021; revised 10 May 2021; accepted 1 June 2021; published online 24 June 2021) [Editor: Stephen Paul Robinson] Pages: 4516–4533


I. INTRODUCTION
When the winds blow above the sea, underwater noise may be created by a variety of mechanisms including windgenerated surface waves, the impact of water droplets, and the entrainment of air bubbles into the surface layer. For most locations in the world's oceans, wind-generated sound is the dominant source of underwater ambient noise (Knudsen et al., 1948) over a broad range of frequencies (400 Hz-50 kHz). However, at low frequencies (5-400 Hz), anthropogenic noise from commercial shipping overtakes wind noise as the dominant source of ambient sound (Wenz, 1962;Ross, 1976;Hildebrand, 2009).
Extensive underwater sound recordings allow an empirical model of wind-generated noise to be developed. Autonomous recordings collected by our group encompass more than one hundred cumulative years of data and capture high wind events. They were collected both on shallow continental shelves and in open ocean deep-water settings. The data are broadband (100 kHz) and with instrumental noise levels low enough to measure contributions of wind noise up to $20 kHz (Wiggins and Hildebrand, 2007). In the frequency band 400 Hz-20 kHz, wind-generated noise often dominates other sources, and a clear correlation is seen between wind speed and noise level. At lower frequencies of 10-400 Hz, local wind speed and noise levels are rarely correlated, and care must be exercised to discern the impact of wind in the absence of other sources of underwater sound. The source mechanisms for wind-generated noise in the band 400 Hz-20 kHz appear to be surface wave and turbulence interactions at low wind speeds (<3.3 m/s) and bubble oscillations at higher wind speeds. The source mechanism for wind-generated low frequency noise (10-400 Hz) is uncertain. Wind-generated noise increases with wind speed, but only up to a point. At the highest wind speeds (u >15 m/s), noise levels begin to decrease for frequencies >10 kHz, owing to interaction between bubbles and screening of noise radiation in the presence of high-density bubble clouds.

A. Physics of underwater noise
Agitation of the sea surface due to the passage of wind is a major source of underwater ambient noise (Wilson, 1980). Interaction of the wind with the sea surface detaches water droplets which both impact the sea surface and entrap underwater bubbles to generate sound (Franz, 1959). Droplet impacts are particularly created when breaking waves occur at increasing wind speed. The character of the sound produced is related to the first impact by the droplet and by the formation and oscillation of an entrapped bubble beneath the surface (Pumphrey and Elmore, 1990;Gillot et al., 2020). The sound impulse created by the first impact is related to the kinetic energy of the droplet, a function of a) Electronic mail: jahildebrand@ucsd.edu, ORCID: 0000-0002-5418-9799. droplet size and impact speed. The sound spectra radiated by the droplet impact has been shown to cover a wide frequency band ($100 Hz-10 kHz; Franz, 1959). The probability of entrapment of an air bubble also depends upon the droplet size and speed at impact (Pumphrey and Elmore, 1990), with some regions of the velocity and droplet sizespace more likely to produce bubble entrapment (regular entrainment), while other conditions may or may not result in bubble formation (irregular entrainment). With regards to noise production from entrapped bubbles there are two separate classes of models: those that consider the response of individual bubbles (Loewen and Melville, 1991) and those that consider the collective oscillations of bubble clouds (Prosperetti, 1988;Means and Heitmeyer, 2001;Tkalich and Chan, 2002). A power law of between two and four is predicted for noise dependence upon wind speed due to bubble oscillations (Evans et al., 1984;Kerman, 1984).
Under low-wind conditions where there may be few or no breaking waves to produce bubbles, individual bubble or bubble cloud models may not adequately explain the full bandwidth of observed noise (2 Hz-50 kHz). Under conditions without breaking waves, three additional mechanisms have been put forward (Kewley et al., 1990) to explain wind-generated noise as a function of wind speed and frequency: (1) wind turbulence (Wilson, 1979), (2) ocean surface wave interactions (Kibblewhite and Ewans, 1985;Webb, 1998), and (3) surface wave turbulence interactions (Yen and Perrone 1979). In the frequency band 10-200 Hz it has been estimated that noise generated by surface waveturbulence interaction may be dominant over the other two mechanisms (Yen and Perrone, 1979;Carey and Browning, 1988), and that noise from surface wave turbulence interaction should be directly proportional to wind speed and decrease as $1/f 2 .
Models that calculate noise are based on placing many random sources near the sea surface and then summing their contributions (Kuperman and Ingenito, 1980). In these models, a plane of monopole noise sources is positioned at onequarter wavelength depth beneath the surface, leading to a dipole radiation pattern that is maximum in the downward direction. Further models have been proposed that account for attenuation of noise by sea surface bubble clouds, based on the distribution of bubbles with wind speed (Weston, 1989). At frequencies above 8 kHz and wind speeds above 15 m/s it has been found experimentally that noise levels decrease with increasing wind speed, likely due to scattering and absorption of sound by near surface bubbles (Farmer and Lemon, 1984).

B. Underwater noise models
Empirical relationships for underwater noise have been developed with wind speed as the primary independent variable (Knudsen et al., 1948;Wenz, 1962;Wilson, 1983). The most general empirical models (Carey and Evans, 2011) represent noise power (N dB re: l Pa 2 /Hz) as a function of frequency (f), observation depth (d), and wind speed (u) as follows: Þþ 20 Ã n f ; d; u ð Þlog 10 u ð Þ À 10 Ã m f ; d; u ð Þlog 10 f ð Þ: (1) This model recognizes that there is an expected logarithmic dependence of noise on wind speed, with the parameter 20*n (f,d,u) giving the slope of the dependence. A value of n ¼ 1 gives noise intensity that varies as the square of the wind speed, which has an intuitive appeal since the wind stress on the sea surface should vary as the square of wind speed. The parameter 10*m (f,d,u) gives the frequency dependence of the noise. Based on Knudsen et al. (1948) and others (Kerman, 1984;Ma et al., 2005), the expected value of m for frequencies above 1 kHz is $5/3 to 2. For frequencies below 400 Hz, m has been found to be nearly zero, with some dependence on wind speed (Urick, 1984;Carey and Evans, 2011). The primary depth-dependence of the noise is expressed in the parameter O(f,d,u). Noise generated at the sea surface will experience attenuation as it propagates to depth (Urick, 1975;Short, 2005;Kurahashi and Gratta, 2008), and the attenuation becomes significant at high frequencies and for deep sensors. An expression for the attenuation of noise at depth d can be obtained by considering the noise level received on an omnidirectional hydrophone J o as follows: (2) expressed as the sum over contributions from the directional source J X of ambient noise intensity per unit solid angle emitted by sea surface noise sources such as bubbles.
Assuming straight-line ray paths and no reflections from the bottom, this becomes where J 1 is the average intensity per unit solid angle radiated by a unit surface area, h is the angle of the ray arriving at the hydrophone, and a is related to the sound absorption coefficient a (Ainslie and McColm, 1998) by ad ¼ À10 log e Àad ð Þ , assuming surface dipole sources (Short, 2005). The depth-dependent noise correction O then becomes (4) which will strongly limit high frequency noise (> $40 kHz) at great depth ($1000 m).

C. Wind models
There have been significant improvements over the past decade in the measurement and modeling of ocean wind using remote sensing data (Bourassa et al., 2019). Ocean surface wind speed can be measured by satellite using microwave radiometers and scatterometers. In the former, a radiative transfer model is used to calculate microwave emission from the ocean surface, as well as absorption and emission by the atmosphere (Meissner and Wentz, 2012). Radiometer sensors that can measure wind speed include the Special Sensor Microwave Imager (SSM/I), Special Sensor Microwave Imager Sounder (SSMIS), Tropical Rainfall Mission Microwave Imager (TMI), Global Precipitation Mission (GMI), Advance Microwave Scanning Radiometer (AMSR), and WindSat. Radar scatterometers capable of wind measurement include the Quik Scatterometer (QuikSCAT), Advanced Scatterometer (ASCAT), Advanced Microwave Scanning Radiometers (AMSR-E and AMSR2), and the Micro-Wave Radiation Imager (MWRI). The radiometer and scatterometer data are validated with in situ measurements, with agreement to within 0.8 m/s (Bourassa et al., 2019). The largest concerns are contamination from rain, issues with calibration at very high wind speeds, and the lack of data near land and over sea ice.
Global models of the wind are constructed by combining satellite winds and in situ observations to produce a gridded dataset. One such model is the Cross-Calibrated Multi-Platform (CCMP) wind analysis that uses a variational analysis method that includes an intercalibration of satellite radiometers and a refined sea-surface emissivity model and radiative transfer function to derive surface winds (Atlas et al., 2011;Wentz, 2016). The European Center for Medium-Range Weather Forecasts (ECMWF) Interim Reanalysis winds are used in the CCMP V2.0 as the initial estimate of the wind field. The CCMP model references all wind observations (satellite and in situ) to a height of 10 meters.

A. Underwater sound dataset
A significant dataset of underwater sound has been collected by our lab over more than a decade using autonomous acoustic recorders . Beginning in 2004, the High-frequency Acoustic Recording Package (HARP) data logger was developed to provide long-term ($1 year) and broadband ($100 kHz) recording capabilities for remote acoustic monitoring (Wiggins and Hildebrand, 2007). The HARP is configured as a seafloor or water-column autonomous mooring with internal data storage and battery power. These instruments have been used for a range of studies of ambient noise and anthropogenic sound sources such as ships and airguns (McKenna et al., 2012;Roth et al., 2012;Wiggins et al., 2016;Gassmann et al., 2017).
A dataset of 291 instrument deployments with 50 455 days (138 years) of acoustic data collection between 2007 and 2019 are examined in this study (Appendix, Table V). Data were collected at 72 unique sites that cover a broad range of latitude and depth, including areas with seasonally strong winds such as the Gulf of Alaska, Western Atlantic/ Gulf of Mexico, and Southern Ocean (Fig. 1). In almost all cases, the instrument package was placed on the seafloor and the hydrophone was positioned 10-30 m above the instrument package.
Calibrated hydrophones and recording electronics are used in the HARP to determine accurate received sound pressure levels. All hydrophone sensors are lab-calibrated before deployment and at the end of service life, and representative hydrophones are full system calibrated at the U.S. Navy's transducer evaluation center, TRANSDEC, in San Diego, CA. The design of the HARP hydrophone employed piezoelectric ceramic sensors, either as single or as multiple elements, to cover the frequency band from 10 Hz-100 kHz (Table VI). When multiple elements were employed, one group was optimized for the low-frequency band (10 Hz-2 kHz or 20 kHz) using a bundle of six individual elements (Benthos AQ-1) and the other was optimized for the high frequency band using a single spherical element (typically ITC 1042). Changes in the HARP hydrophone design over the period of this study were primarily in the amount of gain applied to the high frequency band and the location of the crossover between the high and low frequency sensor bands. For the purpose of this study, we recognize five distinct hydrophone designs that are numerically designated by their pre-amplifier numbers (Table VI). The acoustic data were sampled at 200 kHz (16-bit resolution) and processed into 100-Hz bin-width, 5-s duration spectral averages for the band 100 Hz-100 kHz, and after decimating by a factor of 5, into 10 Hz bin-width spectral averages for the band 10 Hz-1000 Hz. Times when the instrument was writing to its internal disk storage were eliminated from these spectral averages. These 5-s spectral averages were further combined to obtain an hourly estimate of the noise spectra over 10 Hz-100 kHz.  Table V, for listing of deployment locations, dates, and seafloor depths. Sensor depths are 10-30 m above the seafloor.

B. Wind data
The CCMP v2 model was used to estimate the wind speed at 10 m altitude above each HARP deployment location simultaneous with the acoustic recording (Atlas et al., 2011;Wentz, 2016). The CCMP v2 model provides estimates on a 0.25-degree spatial resolution grid with 6-h time resolution. Wind data were used from the closest grid point to the deployment, no more than 0.125-degree (14 km) distant. The wind data were interpolated to a 1-h resolution and combined with the 1-h average of the acoustic data (Fig. 2). The lack of temporal resolution in the wind model precluded incorporating the impact of short-term high-speed wind gusts into the modeling effort. For each deployment, plots were made to assess the correlation of sound levels and wind speed at various frequencies (Fig. 3), and deployments with little or no correlation were removed from the analysis. The Southern Ocean site shown in Fig. 3 is particularly illustrative of the contributions of wind because it is remote from major shipping lanes. Sites that were nearby islands or coastlines with high topographic relief (e.g., Kona, HI) showed poor correlation between the predicted wind and the sound level, presumably due to lack of spatial resolution in the global wind model. Outliers in the wind speed versus noise level plots were removed using a spline fit to the data, eliminating data points with greater than 95% standard deviation from the mean of the fit.
A comparison between the sound data and the wind data at each site was conducted by segmenting the wind data into speed categories corresponding to Beaufort sea-state numbers (WMO, 1970) as given in Table I, and plotting noise versus broadband frequency for each Beaufort number (Fig. 4). These provide an assessment of the shape of the noise spectra versus frequency and allow determination of which portions of the spectral band are wind-related. There is a high correlation between wind speed and noise level in the frequency band 300 Hz-20 kHz. The noise level is flat with frequency, and independent of wind speed for frequencies above $20 kHz, suggesting that these data are set by instrumental electronic noise. At the lowest frequencies (<100 Hz) wind speed dependence is only observed by careful selection of data for high wind events, in areas secluded from distant shipping noise, and after removing periods of local anthropogenic noise. For this dataset, these conditions only occurred for Southern Ocean sites, and for sites that were sheltered from long-range propagation by local bathymetry such as in the California Borderlands and in the Gulf of California.
An alternative approach to understanding the relationship between the noise and wind data is to plot the noise level versus wind speed at a fixed frequency (Fig. 5). At low frequencies (<100 Hz), these plots show little correlation between noise level and wind speed, whereas for frequencies of 300 Hz-20 kHz an approximately linear relationship is observed for wind speed scaled as log 10 . A regression analysis for wind speeds u > 5 m/s was calculated to provide an estimate of the wind dependent parameter n(f,d,u) from Eq. (1) (red lines in Fig. 5). A change in slope is often observed for wind speed u < 5 m/s and a lower value of n(f,d,u) is observed, related to the lack of breaking waves (Evans et al., 1984;Kerman, 1984). There is also a change of slope for the highest frequencies (10 and 20 kHz) at the highest wind speeds, likely related to interference between bubbles (Farmer and Lemon, 1984).   Fig. 3). Red line is linear regression for data above log 10 (5 m/s) ¼ 0.7 (blue datapoints).

C. Model
A model for wind-generated underwater noise was created to match the observed wind and sound data described above following the form of Eq. (1). Starting parameters for the model, especially the wind dependence n(f,d,u) and the frequency dependence m(f,d,u) were obtained from previously published studies (Piggott, 1964;Perrone, 1969;Burgess and Kewley, 1983;Urick, 1984;Kewley et al., 1990;Marrett and Chapman, 1990;Chapman and Cornish, 1993;Ma et al., 2005;McDonald et al., 2008;Carey and Evans, 2011;Reeder et al., 2011). The goodness of fit between the model and the observation data was determined as a function of sensor depth, binning the observation data into 200 m intervals. Where the model systematically deviated from the trends of the observation data, the parameters n(f,d,u), m (f,d,u), and O(f,d,u), were adjusted to allow for better agreement. The goal of these adjustments was to keep the average mis-match between the model and the observation data under $1 dB for all depth bins. The volume of the dataset, the large numbers of free-parameters in the model, and the need to exclude data that were contaminated by anthropogenic noise sources, precluded an automated parameter estimation. Every effort was made to keep the size of both linear and non-linear parameters in the model to a minimum and to test for sensitivity to individual parameters.

A. Wind dependence parameter
The wind dependence parameter n(f,d,u) is obtained for each dataset from linear regression of noise versus wind speed (Fig. 5). The values of n(f,d,u) for the entire dataset are plotted as a function of frequency in Fig. 6 and median numerical values and linear regression goodness of fit are presented in Table II. At low frequencies (<100 Hz) factors other than the wind significantly contribute to underwater ambient noise, resulting in low correlation and wind dependence parameter estimates of 0.2-0.5 (Table II). For frequencies in the range 200 Hz-10 kHz, the wind dependence parameters are in the range n ¼ 0.9-1.2, near the expected value of 1.0 due to wind stress on the ocean surface (albeit with extended tails for the distribution at 200 and 500 Hz, suggesting some vessel traffic contamination). At the highest frequencies that can be discerned above instrumental noise, 20-30 kHz, an alternate phenomenon is manifest, with wind dependence parameters of $0.5. Indeed, at these frequencies, a simple linear regression does not represent the curvilinear relationship between wind speed and noise, as illustrated in Fig. 5.
To account for the depth and frequency dependence observed, additional factors (nfacl, nfacnl, nfacf) were introduced into the wind parameter n(f,d,u) as follows: where d is depth, nfacl is a linear depth parameter (nfacl ¼ 1000 m), nfacnl is a non-linear depth parameter (nfacnl ¼ 400 m), nfacf is the frequency at which the nonlinear depth parameter begins, and An, Bn are constants (Table III). The impact of the linear and non-linear depth terms is to decrease n increasingly with both depth and above frequency nfacf. Wind speed dependence was also introduced into the wind parameter n(f,d,u). At the lowest Beaufort force (numbers 1 and 2) it was determined that n ¼ 0.5, while n ¼ 1.0 at Beaufort force 3, and n ! 1.5 for higher Beaufort force numbers.

B. Frequency dependence parameter
The frequency dependence parameter m(f,d,u) is obtained from plotting noise level versus frequency, binned by Beaufort number, as in Fig. 4. At low frequencies (<100 Hz), noise levels at high wind speed sometimes exceed those of anthropogenic noise sources (e.g., commercial shipping). However, at low wind speeds, the only way to observe wind noise is to select for locations that are distant or shielded from ship noise (McDonald et al., 2008;Reeder et al., 2011) and to select for periods of time that exclude local ships and periods of high instrumental noise from flow or strum (see lowest noise levels in Fig. 4). The frequency parameter m(f,d,u) was found to be À0.5 to þ0.3 for frequencies below 400 Hz. At higher frequencies (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) To account for the depth dependence of m(f,d,u) additional factors (mfacl, mfacf) were introduced as follows: where d is depth, mfacl is a linear depth parameter (mfacl ¼ 1000 m), mfacf is the frequency at which the non-linear depth parameter begins (mfacf ¼ 15 kHz), and Am is a constant (A ¼ 2.5 dB). In practice, the impact of this term is to decrease m(f,d,u) from -1.5 to À2.0 for frequencies above 10 kHz and at shallow depths (<500 m).

C. Offset parameter
The offset parameter O(f,d,u) adjusts the noise model to allow for its continuity over frequency and wind speed with changes in the wind n(f,d,u) and frequency m(f,d,u) parameters. It also allows explicit adjustment for depth-dependent attenuation, as described earlier using the model of Short (2005). An additional depth-dependent term was found to be needed as follows: where d is depth, Ao is an amplitude in dB (Ao ¼ 2.8 dB), and Bo is the depth for 1/e decrease in influence (Bo ¼ 600 m). This term partially accounts for bottom interaction, which was   (2005) analysis, adding about 2 dB at 200 m depth, and 1 dB at 600 m.

D. Hydrophone calibration
Segregating the dataset by hydrophone type allowed testing for systematic deviations from their laboratory calibrations. The average misfit between the model and data for each hydrophone type revealed small (0-4 dB) deviations from the laboratory calibrations. Figure 7 shows the average transfer-function correction by hydrophone type. Most of the deviations from laboratory transfer functions occur near the crossover frequency between lower and upper frequency sensors. For the 500 and 600 series hydrophones, at the crossover frequency near 3-5 kHz, deviations of 2-3 dB are observed. For the 700 series hydrophone, deviation is seen at the crossover frequency near 20 kHz, and there also appears to be a linear correction to the gain with frequency, with an amplitude of about þ4 dB at 20 kHz and À2 dB at 300 Hz. Since these corrections are systematic by hydrophone type, they are applied as a correction to the dataset.

E. Wind-generated noise model
The wind-generated underwater noise model is plotted as a function of frequency for a set of wind speeds or sea-states (Knudsen et al., 1948) as shown for the models in Fig. 8 (left) calculated for three sensor depths (100, 500, and 1000 m). At low wind speed (Beaufort 1-4) noise levels decline from 10 to 100 Hz, and then increase from 100 to 400 Hz to a local maximum at 400 Hz, before falling with slopes of $m ¼ 1-1.5 for frequencies above 1 kHz. For higher wind speeds (Beaufort 5-11) the local noise maxima at 400 Hz disappears and noise decreases uniformly for frequencies between 10 Hz and 1 kHz. At high wind speeds (Beaufort 8-11) and high frequencies (>10 kHz), a striking feature of the model is the crossover of noise curves producing lower noise levels at higher wind speeds for these conditions. For models at depth (1000 m), attenuation becomes a factor at frequencies >10 kHz, resulting in low noise levels independent of wind speeds. The parameters used to calculate these models [n(f,d,u), m(f,d,u)] and O(f,d,u), are given as a function of frequency and Beaufort force in Fig. 9.
The goodness-of-fit for the model is calculated as an average for all the deployments in each 200 m depth interval (Fig. 10). All depth interval averages fit the noise model to within 61 dB across the frequency band 200 Hz-20 kHz, except for the shallow interval 0-200 m. When all the deployments are averaged together, the misfit is <0.25 dB. The 0-200 m depth interval shows systematic deviations of $1-1.5 dB with frequency, presumed to be due to bottom interactions and shallow water propagation effects (Ingenito and Wolf, 1989) that create constructive and destructive interference that is not included in the model. Figure 4 plots the model prediction (dashed red line) against the data collected at the Southern Ocean site. The model and Southern Ocean average noise data (dashed black line) agree well in the frequency range 200 Hz-20 kHz, but deviate for frequencies >20 kHz due to instrumental electronic noise. Below 200 Hz, the model and data agree only during the lowest noise periods, presumably due to the influence of residual anthropogenic sources and instrumental noise from periods of high flow and strum.

V. DISCUSSION
A comparison of the present model to previous noise models and observations is given in Table IV. The previous studies are divided into shallow (< 200 m) and deep-water ($1000 m) with the present model calculated at 100 and 1000 m for comparison. As noted previously by Urick (1984) the observational data are inconsistent between published sources, presumably due to differences in local conditions as well as system calibrations and data analysis methods. For the shallow water data, the pioneering study of wind noise by Knudsen et al. (1948) has comparable sound levels to the present model [root-mean-square (rms) difference of 1.6 dB], but the non-linearity of noise with wind speed suggested by Knudsen et al. (1948) between Beaufort Force 2-5 is not substantiated by the present model. The wind speed curves of Wenz (1962), adjusted for shallow water, are consistent with both the Knudsen et al. (1948) model and the present model, except at the lowest wind speeds (Beaufort force 1). The data of Piggott (1964) and Wille and Geyer (1984) are both higher in level (5.5 and 2.9 dB rms, respectively) than the present study. The more recent shallow water studies of Ma et al. (2005) and Nystuen et al. (2010) are in good agreement with the present model, except for a lower than expected value at low wind speed (Beaufort force 2). For deep water, the present model was calculated at a sensor depth of 1000 m for comparison with previous studies. Following installation of US Navy surveillance arrays beginning in the 1950s, studies were conducted to document the wind speed dependence of ambient noise in deep water. During commissioning of these arrays in the Atlantic, Ross (1954) produced a generalized ambient noise spectra (see the supplementary material Fig. 1) 1 that is largely in agreement with the present model (1.1 dB rms difference). Subsequent studies conducted by Wenz (1962) and Perrone (1969) are lower than the present model (4.2 and 5.0 dB rms, respectively) especially at the low (Beaufort force 1) and high (Beaufort force 8) ends of the wind spectrum. This may be the result of a lack of accurate wind speed data, and/or use of Atlas of Climatic Charts of the Oceans or similar to approximate wind speed; presumably good estimates were available for average wind speeds, but poor estimates for periods of very low and very high winds. The idealized ambient noise spectra of Urick (1984), based on both a literature review and expert knowledge of spectral behavior of noise sources, is consistent with the present model (1.4 dB rms difference), as are the measurements of Cato (1976) made in waters near Australia, and the measurements of Reeder et al. (2011) made in the Bahamas.
Several features of the empirical noise model are noteworthy and provide insight into possible source mechanisms and areas where more understanding is needed. The first point is that the model at low frequency (10-100 Hz) separates wind contributions to ambient noise from those of anthropogenic sources such as commercial shipping. For most locations, omni-directional ambient noise at low frequencies is dominated by the sum of many distant ships, or in some locations seismic surveys (Wiggins et al., 2016), and the contribution of wind is not easily discernable. Many efforts to separate wind noise from anthropogenic noise have been based on the use of an acoustic array to separate distant (horizontally propagating) from local (vertically propagating) sources (Kewley et al., 1990;Chapman and Cornish, 1993;Farrokhrooz et al., 2017). Instead, the model we present is derived from omni-directional hydrophones placed at locations sheltered from ship noise at low wind speeds (McDonald et al., 2008;Reeder et al., 2011), and at times of extreme wind speed when ship noise may no longer be dominant. Sites in the southern hemisphere are particularly helpful in minimizing the presence of distant shipping noise.
The low frequency portion of the model presented here has frequency dependence of about 5 dB/decade (m ¼ -0.5) or less for noise below 400 Hz. This is in contrast to the slopes of$15 dB/decade (m ¼ -1.5) seen above 1 kHz in the model. Both bubble cloud and surface wave-turbulence source mechanisms (Yen and Perrone, 1979) have a presumed 20 dB/decade frequency dependence (1/f 2 ). The portion of the noise model above 1 kHz is roughly in accord with this slope, but the noise model below 400 Hz is not, pointing out an incomplete understanding of what is the source of low frequency wind-related noise in the absence of shipping. Along these lines, the frequency band between 100 and 400 Hz may be a zone of transition between two different noise generation mechanisms. The presence of flow noise and cable strum is a possibility at low frequencies. Care has been taken in the hydrophone design and cable attachment to minimize these effects, but they are difficult to protect against during periods of high flow. However, the discontinuity in low frequency, low wind-speed noise makes it less than expected, so additional hydrophone flow and strum noise would not explain the difference.
For low wind speeds (u < 3.3 m/s) noise levels and wind speed have a directly linear relationship (n ¼ 0.5). Whereas for higher wind speeds (u > 5.5 m/s), the relationship is a power law of two or three (n ¼ 1.0 to 1.5). This suggests that there is a transition between a source mechanism of surface waveturbulence interactions at low wind speeds, with a linear dependence on wind speed, and the presence of breaking waves and their associated bubble formation at higher wind speeds, with a higher power law relationship.
Another feature of the noise model is that at frequencies >10 kHz and for Beaufort number > 7 (u > 17 m/s), noise levels diminish with increasing wind speed, as shown for the Southern Ocean data in Fig. 3. Given that bubble cloud oscillations appear to be the dominant source mechanisms under these conditions, one possibility is that strongly energetic winds inject greater volumes of bubbles and to a greater depth, such that there may be interference between bubbles either in limiting additional sound generation and/or in limiting propagation of the generated sound from the near surface source region. It has been previously suggested that the acoustic radiation from newly formed bubbles can be both scattered and absorbed by previously entrained bubbles (Farmer and Lemon, 1984;Updegraff and Anderson, 1991).
Despite the smooth appearance of the noise model in both frequency and in wind speed (Fig. 8), the parameters used to generate it are discontinuous in both variables (Fig. 9). This is a result of the piecewise construction of the model, resulting in discontinuities in the wind dependence and frequency dependence parameters, and associated discontinuities in the offset parameter. A full inversion for these parameters may allow for the development of both a smooth model and a smooth set of model parameters, and help to discover more subtle features of the dataset than possible with the current approach. Also, despite the large volume of data that went into the current model, it will be possible to validate and update the model with more recent wind and acoustic data as they become available.

VI. CONCLUSION
A large underwater noise dataset and global wind model were combined to produce an empirical model for the dependence of underwater noise on local winds. The model explicitly separates noise generated by sources related to the wind from anthropogenic sources of underwater sound. The TABLE IV. Comparison of 1 kHz wind-noise spectrum level (dB re: lPa 2 /Hz) between the model presented here and that of previous ocean noise models and data. The values for previous studies are given with 0.5 dB precision due to the difficulty of estimation from published graphics. The rms (dB) difference between the present model and the published study is given in the right-hand column, comparing the model with the sensor depth at 100 m and shallow water studies, and the model at 1000 m and deep water studies. model for wind-generated noise shows small (5 dB/decade) frequency dependence at low frequencies (10-100 Hz), and larger ($15 dB/decade) frequency dependence at higher frequencies (400 Hz-20 kHz). The relation between noise level and wind speed is linear for low wind speeds (u < 5 m/s) and increases to a higher power law (two or three) at higher wind speeds, suggesting a transition between the presence of breaking waves and their associated bubble formation. At the highest wind speeds (u > 15 m/s), noise levels begin to decrease at high frequencies (>10 kHz) due to interaction between bubbles and screening of noise radiation in the presence of high-density bubble clouds.

ACKNOWLEDGMENTS
The authors acknowledge the long-term support of a large number of agencies and individuals in the collection of these acoustic data including, the Chief of Naval