The hull shape is clearly important when resolving short wave components. The main drivers are the stability of the buoy. We typically want to have the top of the buoy stay above the water surface, in particular for GPS acquisitions and radio transmission. We also wish to avoid rotation of the buoy relative to the water around it, and finally the buoy has to be big enough to be visible for recovery and small enough to be easily handled and to follow the motion of short waves. One final driver is the overall cost of the buoy. Because they also measure whitecaps with a camera and turbulence in the water, the SWIFT buoys use a spar shape that is 1.8 m tall. Such a shape is not ideal for short wave measurements because it resonates for heave excitation at a frequency around 0.8 Hz.

With all these constraints in mind we found that a nearly spherical shape with ribs and an additional skirt provided a good water-following behavior, whereas spherical shapes performed more poorly. Three-dimensional printing was tested without much success due to the porosity of the printed material. For the small number of buoys that we needed we finally settled on glass spheres, for which we had other oceanographic uses for buoyancy in deep water moorings. The standard ribbed cage for these spheres (Figs.  and ) was augmented by a 3 cm wide skirt, as shown in Fig. , providing a nice water-following capability.

The accelerometer and the GPS system are directly integrated in a general-purpose oceanographic Advanced Low Energy Electronic System (ALEES) board developed by Ifremer “Unité Recherche et Développements Technologiques” (RDT) especially for autonomous applications that need very low power consumption. This generic board uses a 32 bit microcontroller working at 48 MHz, a 1 Mb flash memory and 128 Kb RAM. The data are stored in a standard micro secure digital high-capacity (SDHC) memory card. The GPS and the accelerometer acquisitions are not synchronized and the acquisition rates are 1 and 25 Hz, respectively.

The integrated accelerometer is a STMicroelectronics model LIS3DH (when used with this configuration, this will be referenced as the SKIB STM buoy version), already incorporated in the ALEES board for other uses, namely the detection of strong motions for underwater sensors. This low-cost (less than USD 2) component was chosen for its very low power use: between 2 and 6 µA at 2.5 V.

A specific board was designed to control the GPS acquisition and send the buoy position via the Iridium board. We typically programmed the buoy to send position messages every 10 min, in order to be able to find the instruments at sea in highly variable currents. The ALEES and GPS boards can be controlled by a Zigbee wireless link. This wireless link also allows the user to set up the buoy and to recover the data without opening the glass spheres, allowing the system to power on and off.

All the system, including the electronic boards, battery pack and antennas (Zigbee, GPS, Iridium) are mounted inside a 25.4 cm diameter glass sphere, which is vacuum-sealed (see Fig. ). Standard prices for all the parts in the year 2015 were about EUR 1100 for all electronics, half of which is for the Iridium and GPS equipment and another EUR 1100 of which is for the hull and mounts inside of the hull. That expensive choice of the hull was, in our case, justified by a possible reuse for other oceanographic applications.

For a detailed validation we have also integrated a more accurate sensor in two of the SKIB buoys (this is now referred to as the SKIB SBG buoy version). In those buoys an SBG Ellipse inertial measurement unit (IMU) was used, set to an acquisition rate of 50 Hz. However, this sensor significantly increases the equipment cost and power consumption, with a unit price typically above USD 4000.

Buoy testing started with the verification of the expected acceleration accuracy in a wave tank, followed by a comparison with in situ measurements with a reference wave buoy.

The laboratory tests were very useful for testing various hull shapes, from spheres to short cylinders. These led to the addition of the plastic skirt that effectively removes rotations around the horizontal axes, with a limited impact on the water-following capacity for short wave components. This final design has a heave transfer function of almost 1 up to 0.8 Hz, decreasing to 0.6 at 1 Hz, as established in wave basin tests . This extends the useful range of buoys such as Datawell Waveriders or SWIFTs to a high frequency. For in situ validation, the SKIB buoy was deployed drifting within 200 m of a MkIII Datawell Directional Waverider of 70 cm diameter, moored in a region of weak currents with a mean water depth of 60 m at 48.2857∘ N, 4.9684∘ W. This Waverider buoy is part of the permanent CEREMA wave buoy network, with the World Meteorological Organization number 62069 . This buoy provides measurements of the first five moments for frequencies of 0.025 to 0.580 Hz, based on accelerometer data.

Contrary to who strapped their new acquisition system on a Waverider buoy, we wanted to validate the full system, including the hull response. As a result the different sensors do not measure the same waves (with the same phases) but should be measuring the same sea state, i.e., the same spectrum, moments and derived parameters.

The test presented here was performed on 21 September 2016, from 10:44 to 11:56 UTC, following a similar test in 2015 with only a SKIB and a with a different GPS receiver but the same hull and a Datawell Waverider. The results were very similar. In the 2016 experiment, we also deployed a SWIFT buoy and a ship-mounted stereo-video wave system . Pictures of all these systems as used during the experiment are shown in Fig. .

The SWIFT model used is shown in the water in Fig. c. It uses a GPS receiver integrated with an IMU (Microstrain 3DM-GX3-35), a Doppler velocity profiler (Nortek AquadoppHR), an autonomous meteorological station ultrasonic anemometer (AirMar PB200), a digital video recorder system and a real-time tracked radio frequency transmitter. The wave spectra for each 10 min burst are calculated as the ensemble average of the fast Fourier transform of 16 sub-windows with 50 % overlap, which results in 32 degrees of freedom and a frequency bandwidth df=0.0117 Hz in the range 0.05<f<0.5. The IMU data give information about the tilt and horizontal rotation, as well as accelerations, of the SWIFT as it follows wave motions. Note that the hull shape of the SWIFT follows displacements and velocities at the sea surface but not surface slopes. Hence, only velocities and accelerations are used in wave processing. Post-processing of the merged GPS and IMU data is applied as a classic RC (resistor–capacitor) filter to exclude signals at frequencies lower than f<0.04 Hz.

The stereo-video system is the same as used by , based on a pair of synchronized video cameras (2048×2456 pixels) BM-500GE JAI, mounted with wide-angle lenses. Here the system was installed at the bow of R/V Thalia, a 24.5 m ship of the French coastal oceanographic fleet (Fig. d). The cameras are located approximately 7 m above the sea surface, and an Ellipse-D inertial measurement unit is fixed on the bar joining the cameras to correct for ship motion with 0.2∘ accuracy on all rotation angles. The video processing follows . The only difference in the present case is that the mean surface plane correction, which was used for deployment from fixed platforms, is replaced by an optimization of the rotation matrix given by an SBG motion package mounted with the cameras. The resulting surface elevation maps ζ(x,y,t) acquired over 30 min records at 12 Hz are gridded in a 10 by 10 m square surface with 0.1 m resolution. This square moves with the mean velocity Um, relative to the solid Earth, as given by the GPS data. The 3-D spectrum E(kx,ky,f) was obtained after applying a Hann window in all three dimensions to the elevation maps over time intervals of 85.33 s (1024 frames), with 50 % overlapping as well. As a result, the energy over frequency and wavenumber are in a reference frame moving at the speed Um, and the measured radian frequency of the waves σm must be corrected by the mean ship velocity (Um) over each time window. So the absolute frequency in an Earth reference frame is ω=σm-k⋅Um. This procedure is particularly prone to errors for wave components longer than 20 m, which are not resolved in the field of view. These longer components can be treated separately using a slope array estimation of the directional spectrum e.g.,, but we focus here on the short waves. The stereo heave frequency spectrum E(f) is obtained by integration over wavenumbers, and it is expressed in terms of the absolute frequency fa, with ω=2πfa=σ+k⋅U, where U is a current field that can be estimated using the drifting buoys.

A comparison of the different sensors at the same sea state conditions is shown in Fig. . For this comparison the records from each sensor were synchronized over 10 min intervals and averaged over the 30 min of the Waverider records and an integration interval from 0.06 to 0.58 Hz. Figure a shows the buoys' drift trajectories for the 1 h of the acquisition, with one color symbol every 10 min and the track of R/V Thalia. The stereo-video record is 20 min, starting at the same time as SKIB and SWIFT acquisitions. The Waverider data correspond to two acquisition of 28 min each, ending at 10:30 and 11:00 UTC.

A closer look at the heave spectra (Fig. b) shows a good correspondence between Datawell Waverider and SKIB buoys at the peak of the spectrum. The main source of error in the SWIFT data, around the peak of the spectrum, was associated with a high-pass filter applied to the IMU acceleration before each time integration. This part of the SWIFT processing, to obtain E(f), was optimized by to reduce the low-frequency noise and to have best agreement with a Datawell Waverider at Ocean Station Papa. This was obtained from the double time integration of the IMU acceleration, with a high-pass filter at each integration, to reconstruct a wave-resolved time series of sea surface elevations. This generally improves the estimation of the spectrum for f>0.1 Hz, but it reduces the level of lower frequencies (as it is also observed in Fig. b).

For the SKIBs we have not filtered the acceleration data and the E(f) was directly obtained from the double time integration of the accelerometer data. Other differences are found for the main direction (Fig. d), which is better retrieved with the SBG IMU. The main benefit of the SBG IMU is the reduction of the noise floor at low frequencies compared to an estimation of the motion from GPS alone. This is most important for swells of long periods and low heights but not critical for our investigation of wind seas interacting with currents.

Figure c and d present the estimates of sθ and θ‾ based on the first- and second-order angular moments. We see a significant difference in the wave spread and mean direction estimates, especially in the first-order estimation (sθ1 and θ1‾). This occurs because the accelerometer is not internally synchronized with the GPS and because they have different characteristics errors. The second-order moment depends only on the horizontal displacements while the first-order moment depends on both horizontal and vertical displacements. So, the second-order moments are more accurate because there is no cross product between different sensors. Although there are differences, the results suggest that the usage of the combination of GPS drifter displacement and vertical acceleration produces a good estimation of the spectrum directionality. These results are particularly important, as the drifter was not equipped with a compass and only used a low-cost GPS receiver. Because the GPS acquisition was limited to 1 Hz in the SKIB with STM accelerometer, the directional analyses are limited to 0.5 Hz (and 0.8 Hz for SBG, which uses only accelerometer data).

A comparison of the different sensors in terms of the usual sea state parameters is shown in Table , with the significant wave height Hs, mean wave period Tm0,1 and peak wave period Tp. Since the only reliable directions are provided by the GPS data alone, we define the peak wave direction Dp and the mean wave direction (θm,2) from the second moments as θm,2=0.5tan⁡-1(B2/A2) with A2=∫E(f)a2(f)df,B2=∫E(f)b2(f)df.

As reported in Table , SKIB results generally agree on Hs, Tm01 and mean directions, with confidence intervals for Hs overlapping with the reference Waverider buoy.

The largest differences are between the SKIB STM and SWIFT buoys and are associated with the filtering of low-frequency content in the SWIFT processing chain (Fig. b) and unfiltered low-frequency noise in the SKIB STM. However, for frequencies from 0.1 to 0.5 Hz, the spectra are consistent with the stereo video and Datawell Waverider data. At higher frequencies, for which we do not have other validation data, the power spectra result follows the same trend and appears realistic up to at least 0.8 Hz, consistent with laboratory tests .

In order to validate the SKIB buoys in different sea state conditions, other deployments were performed next to the buoy 62069: one on 5 August 2015 and another four between 21 and 27 September of 2016. For the 2016 experiment, we used two SKIB buoys equipped with SBG IMU and two others with the STM accelerometer. For the 2015 experiment we had only one buoy equipped with the STM accelerometer. Results for integral parameters are presented in Fig. , and a selection of two spectra with different shapes is shown in Fig. .

For most sea states Hs and Dp are measured correctly (Fig. a and d), with RMSE around 0.3 m and 5.3∘, respectively. As expected, the SKIB SBG agrees best with the Waverider for all the analyzed parameters, and the regression lines for the SBG data (Fig. ) are closer to the ideal correlation line (gray dashed lines Fig. ) than those from the SKIB STM data. In general, the STM accelerometer has more energy at the lower frequencies and this can produce overestimations in Hs and Tm01 measurements.

These errors are confined to frequencies below 0.12 Hz. The main difference between the SKIB and Datawell Waverider was found at the peak wave period (Tp, Fig. c). Higher errors in the identification of the peak frequency are expected a priori as the buoys present different spectral resolutions and different numbers of degrees of freedom . Again the SKIB SBG performs better than SKIB STM.

The buoy's low-frequency noise varies according to the sea state conditions. The errors associated with the low-frequency limit and at the spectrum peak are illustrated in Fig. .

In most sea states analyzed here, the SKIB buoy correctly measured the sea state condition at frequencies higher than 0.07 Hz. In terms of Hs, the instrument usually presents a root mean square error within the statistical uncertainty expected for two perfect devices measuring the same random wave field Eq. 6. However, because of a significant low-frequency noise in SKIB STM we reduced the integration interval for this buoy. The low-frequency noise was reduced by using the SBG IMU, which presented the best performance among the sensors tested here. In summary, we had a good performance of SKIB for f>0.07 Hz, which makes it appropriate to use in the investigation of young wind waves interacting with currents.