## 1. Introduction

Measurements at the air–sea interface are crucial for monitoring, parameterizing, and understanding the behavior of both the ocean and atmosphere. Maintaining sensors here, however, is often difficult: vibration and impact from the constant wave motion is destructive, intermittent saltwater immersion is corrosive, biological fouling is difficult to prevent, and vandalism is common. Measurements of air–sea properties using sensors away from the interface overcome many of these difficulties; satellite-based wind speed (Risien and Chelton 2008) and sea surface temperature (Reynolds et al. 2002) measurements are prominent examples. Such remote sensing of air–sea properties is also possible from beneath the surface, for example, passive acoustic measurements of wind and rain (Ma et al. 2005). The long-term arrays of floats, drifters, and gliders (see special issue of *Limnology* *and Oceanography,* 2008, Vol. 53, No. 5, part 2) offer many opportunities for deploying such measurements, both for large-scale monitoring and as part of localized process experiments.

Pressure variations at fixed points are commonly used to measure surface waves in shallow water. D’Asaro et al. (1996, his Fig. 4) describes estimates of the surface wave field from pressure measurements made on a subsurface float. This paper further develops this approach using a much larger dataset and tests it against standard surface wave measurements from a Waverider buoy. The novelty in this approach is that it exploits the unique properties of surface waves when measured in a Lagrangian reference frame. Measurements are made from a “Lagrangian float” (D’Asaro 2003), an approximately 1-m long, 50-L displacement cylinder with active buoyancy control that matches its density to that of seawater to an accuracy of a few grams (Fig. 1). This small residual buoyancy acting on an approximately 1-m^{2} area horizontal drogue, results in a net vertical motion relative to the water of a few millimeters per second (D’Asaro et al. 2014). Accordingly, the float accurately follows the three-dimensional motion of water parcels at surface wave frequencies.

The pressure along Lagrangian trajectories due to linear, freely propagating, inviscid surface waves is constant. This is apparent at the ocean surface for a floating object; it both follows the water and remains at atmospheric pressure. Appendix A shows that pressure along Lagrangian trajectories is constant to at least second order in wave height at all depths. The increased hydrostatic pressure beneath the wave crests is exactly balanced by the nonhydrostatic acceleration of water trajectories curving downward around the crest. A pressure sensor at the center of a perfectly Lagrangian float thus sees no signal due to surface waves, a feature that allows the float to accurately filter out the very large Eulerian pressure signals (typically equivalent to few meters) due to surface waves and thus measure the much smaller pressure changes due to its own advection by boundary layer turbulence (typically

*ω*and wavenumber

*k*based on

*g*, the acceleration of gravity. The cyclic frequency is expressed as

*z*, positive upward) pressure gradient along a Lagrangian trajectory centered at position

*ρ*is the water density), so pressure has units of meters and the pressure gradient is dimensionless (meters per meter). The float measures pressure fluctuations using

*k*computed from the known frequency and the angle brackets

In practice, this calculation is complicated by a variety of experimental issues that are the focus of this paper. Some readers may wish to skip to section 3, where the results are presented.

## 2. Measurements

The Lagrangian float was air-deployed near Ocean Weather Station *Papa* (OWS-*P*) by the 53rd Air Force Reserve unit, “Hurricane Hunters,” on 9 February 2011, was operated for 100 days, and was recovered by the Canadian Coast Guard Ship (CCGS) *John P. Tully* as part of its annual June cruise to OWS-*P*. During “Lagrangian drifts” occupying most of each day, the float tracked the three-dimensional motion of water parcels, actively controlling its buoyancy to match that of the mixed layer and opening a horizontal drogue to increase its vertical drag (D’Asaro et al. 1996; Harcourt and D’Asaro 2010). The float was repeatedly carried across the mixed layer by the turbulence within the layer at typical vertical speeds of

Pressure was measured at three points along the float hull (red in Fig. 1) using Druck PDCR pressure sensors. The top and bottom sensors were nearly vertically aligned (a lateral separation of about 9 mm) and had a larger vertical separation than pairs including the middle sensor. This pair was thus used in all of the subsequent analysis. All three pressure sensors were sampled at 1 Hz using a 24-bit analog-to-digital (A/D) on a centrally located board. Analog voltage signals from each pressure sensor were transmitted down a shielded twisted pair to this board. Interference from the many other electronic systems was minimized by careful attention to additional shielding and cable location; nevertheless, such signals appear to limit the accuracy of the pressure measurements. There may be additional signals in the pressure that are associated with float motion, float tilts, or flow turbulence, but these cannot be diagnosed using the sensors on these floats. All signals that do not appear to be surface waves are thus called “noise.” This noise was dominated by harmonics of the main 30-s float sampling period. This signal was removed by Fourier transforming the data, replacing values within 0.0018 Hz of each harmonic with random Gaussian complex numbers with an rms amplitude matching that between 0.0018 and 0.0036 Hz of the harmonic, and inverse Fourier transforming. The remaining noise was approximately white with a level of 1–10 mm depending on the sensor.

Wave spectral data were collected at OWS-*P* using a 0.9-m Datawell directional Waverider (DWR MKIII) buoy moored in 4255-m water depth at

The float was deployed about 1.5 km from the Waverider. It drifted southward and westward, reaching a maximum distance of 116 km on day 104, and then moving back to reach a distance of 58 km at the end of the record (day 141).

## 3. Analysis

### a. Simulations

The vertical motion of the float through the broadband, vertically decaying surface wave signal makes the quantitative computation of wave spectra from pressure gradient much more complicated than that in (4). The analysis methods were therefore developed using a simulated set of measurements. A model wave spectrum was defined as having a slope of

Sampling by a moving float was simulated using a sinusoidal float trajectory across a mixed layer of depth *H* with the maximum vertical speed specified. Typical parameter values are *H* = 30 m and

### b. Spectrogram

*t*(s) and radial frequency

*T*is an important analysis parameter. Typical values of 120 s can resolve the surface wave peak frequency while minimizing the effect of float motion. The spectrogram computed directly from the measured pressure will be denoted as

### c. Noise correction (NC)

### d. Depth correction (ZC)

*Z*is the pressure (depth) at the center of the float at the center of the time interval for each spectrogram value. Thus, in Fig. 3, the pressure gradient measured at the float (orange) is much less than that at the surface (blue or red, bottom). The effect is largest for waves with the largest

*k*, that is, the highest frequencies. For a 2-Hz wave,

The ZC correction typically brings

### e. Vertical motion correction (WC)

A float with vertical velocity *W* moves a distance *W* and derives highly accurate corrections for both rectangular and Hanning windows, which depend only on the value of

### f. Acceleration editing (AC)

The value of *W* is computed from the centered difference of float pressure averaged over each spectrogram window. The average acceleration *A* is similarly computed from the difference of *W*. If the fractional change in *W* over the window (i.e.,

### g. Sampling interval correction (dTC)

The WC correction is derived assuming continuous sampling. In this case WC is highly accurate (Fig. 6, red). However, for a realistic sampling interval *T*, we assume that the additional correction depends only on ^{−1}), this predicts the simulated dTC correction to much better than 1% (Fig. 6). Applying this to the simulations (Fig. 4, cyan) removes almost all of the remaining high-frequency bias in the WC correction. The spectrogram

### h. Spectral spreading correction (SC)

A Hanning window spreads energy at

### i. Results

These simulations show that measuring surface waves from a vertically moving platform is much more complicated than measuring from a fixed platform. The analysis finds and corrects for six different sources of error; there are certainly more. Nevertheless, for the cases considered the fully corrected spectrum,

## 4. Application to OWS-*P* data

The methods developed in section 3 are applied to the OWS-*P* data described in section 2. Float data were cleaned as described in section 2. Values of

### a. Sample spectra

Figure 7 shows three sample spectra selected for variable energy levels and the peak wave period. The most energetic spectrum (blue) has a significant wave height of

The least energetic spectrum (black) has a significant wave height of

The intermediate spectrum (red) has a significant wave height of

### b. Statistical comparisons

Significant wave height

The resulting

*β*is a constant,

*P*to estimate

The resulting estimates of ^{−1}. Most of the low values of

## 5. Discussion and summary

The main result of this analysis is that surface waves can be measured from a moving subsurface float by compensating for a variety of effects due to the float’s depth and motion. There are two key effects. First, if the float is at depth *Z*, it cannot measure waves with large values of *W*, the measured wave amplitude is not constant; this introduces a bias that depends on the nondimensional parameter *T*. Large values of *T*. For sufficiently large values, the wave amplitude is small for most of *T* and, in the presence of noise, cannot be measured. Much of this paper is concerned with the details of correcting for this effect. Qualitatively, it introduces an intrinsic trade-off between vertical velocity and spectral resolution. If the float is moving more rapidly, a smaller value of *T* is required; this reduces the spectral resolution *P* data analyzed here span a large fraction of the oceanic range of wind, waves, and mixed layer depths. Useful spectra were obtained during the entire record.

The floats used here measured the waves from the pressure gradient along the floats. This was convenient, since the wave measurements could also provide redundancy for an operationally crucial measurement. However, for wave measurements, the pressure sensors were operating at the limits of their accuracy and were thus quite noisy. Furthermore, the nonhydrostatic vertical pressure gradient is the quantity that causes vertical accelerations

Although this paper addressed the ability of water-following floats to measure surface waves, similar methods should apply to measurements on other subsurface moving platforms, such as gliders and profiling floats. Their subsurface accelerations will follow surface wave trajectories in a manner similar to that of Lagrangian floats, perhaps more closely than their motions when surfaced. Pressure and/or acceleration measurements on such platforms could provide ground truth for satellite-based measurements and operational wave models.

## Acknowledgments

This work was supported by the National Science Foundation (OCE0850551). We are grateful to the people and aircraft of the 53rd Air Force Reserve squadron, “Hurricane Hunters,” for float deployment and to the CCGS *John P. Tully* for recovery. None of this work could be possible without the excellent technical assistance of the Ocean Engineering Department at APL-UW, especially Mike Ohmart and Mike Kenney.

## APPENDIX A

### Wave Properties in a Lagrangian Frame

#### a. Basic equations

*g*is gravity. The velocity potential (Phillips 1977, section 3.2.4) is

*z*is the vertical coordinate and positive up. Velocity

#### b. Lagrangian pressure

#### c. Lagrangian pressure gradients

#### d. Summary

Along a Lagrangian trajectory of a single surface wave to second order:

there are no pressure fluctuations; thus, a float filters surface waves accurately.

the pressure gradient is the same as the Eulerian gradient; thus, a float can measure the surface waves from this gradient.

a vertical pressure gradient proportional to wave slope squared appears, of twice the magnitude and of opposite sign of the mean wave-induced pressure gradient; thus, a float will see a pressure and pressure gradient associated with a changing wave amplitude.

## APPENDIX B

### Wave Measurement Bias Due to Slow Vertical Motion

#### a. The problem

*P*along a Lagrangian trajectory

*W*,

*T*. All frequencies are assumed to be multiples of the lowest resolved frequency, as for the output of an FFT,

*W*, the wave is no longer sinusoidal. Simulating this in Fig. 6 shows the spectral height increasing because of the exponential decay. We wish to analyze this and derive a formula to compensate for it.

#### b. No spectral window

*W*,

*m*, we find the gain

*G*of the Fourier integral—that is, the ratio of output to input spectral level—is

#### c. Spectral window

*h*. Changing to

#### d. Summary

Sampling a wave at a point moving vertically at speed

*W*can significantly bias the amplitude of the measured wave if the distance moved during a sampling interval*T*is comparable to the wavelength. The relevant nondimensional parameter is . The answer is also affected by the FFT window used. For an arbitrary window, the bias is a function only of

. Expressions (B7) and (B12) describe the response for rectangular and Hanning windows, respectively.

These results may not apply for windows that vary rapidly.

## APPENDIX C

### Wave Depth Correction with Spectral Spreading

*Z*to the surface is

*T*, energy is spread over a band of approximately

*Z*and

## REFERENCES

D’Asaro, E. A., 2003: Performance of autonomous Lagrangian floats.

,*J. Atmos. Oceanic Technol.***20**, 896–911, doi:10.1175/1520-0426(2003)020<0896:POALF>2.0.CO;2.D’Asaro, E. A., Farmer D. M. , Osse J. T. , and Dairiki G. T. , 1996: A Lagrangian float.

,*J. Atmos. Oceanic Technol.***13**, 1230–1246, doi:10.1175/1520-0426(1996)013<1230:ALF>2.0.CO;2.D’Asaro, E. A., Thomson J. , Shcherbina A. Y. , Harcourt R. R. , Cronin M. F. , Hemer M. A. , and Fox-Kemper B. , 2014: Quantifying upper ocean turbulence driven by surface waves.

,*Geophys. Res. Lett.***41**, 102–107, doi:10.1002/2013GL058193.Harcourt, R. R., and D’Asaro E. A. , 2010: Measurement of vertical kinetic energy and vertical velocity skewness in oceanic boundary layers by imperfectly Lagrangian floats.

,*J. Atmos. Oceanic Technol.***27**, 1918–1935, doi:10.1175/2010JTECHO731.1.Ma, B. B., Nystuen J. A. , and Lien R.-C. , 2005: Prediction of underwater sound levels from rain and wind.

,*J. Acoust. Soc. Amer.***117**, 3555–3565, doi:10.1121/1.1910283.Phillips, O. M., 1977:

*The Dynamics of the Upper Ocean.*2nd ed. Cambridge University Press, 344 pp.Phillips, O. M., 1985: Spectral and statistical properties of the equilibrium range in wind-generated gravity waves.

,*J. Fluid Mech.***156**, 505–531, doi:10.1017/S0022112085002221.Reynolds, R. W., Rayner N. A. , Smith T. M. , Stokes D. C. , and Wang W. , 2002: An improved in situ and satellite SST analysis for climate.

,*J. Climate***15**, 1609–1625, doi:10.1175/1520-0442(2002)015<1609:AIISAS>2.0.CO;2.Risien, C., and Chelton D. , 2008: A global climatology of surface wind and wind stress fields from eight years of QuikSCAT scatterometer data.

,*J. Phys. Oceanogr.***38**, 2379–2413, doi:10.1175/2008JPO3881.1.Thomson, J., D’Asaro E. A. , Cronin M. F. , Rogers W. E. , Harcourt R. R. , and Shcherbina A. , 2013: Waves and the equilibrium range at Ocean Weather Station P.

,*J. Geophys. Res. Oceans***118**, 5951–5962, doi:10.1002/2013JC008837.Toba, Y., 1973: Local balance in the air-sea boundary processes.

,*J. Oceanogr. Soc. Japan***29**, 209–220, doi:10.1007/BF02108528.