# SFY — A lightweight, high-frequency and phase-resolving wave-buoy for
coastal waters

###### Abstract

Small lightweight wave buoys, SFYs, designed to operate near the coast, have
been developed. The buoys are designed to record and transmit the full time
series of surface acceleration at $52$ Hz. The buoy uses the cellular network
to transfer data and position (up to $80$ km from the base station). This
reduces costs and increases band-width. The low cost and low weight permits
the buoys be deployed easily, and in arrays in areas where satellite and wave
models struggle to resolve wave and current interaction. The buoys are tested
in a wave-flume, the open water and in the breaking waves of the surf. The
conditions range from calm to significant wave heights exceeding $7$ m and
crashing breakers with accelerations exceeding $10$ g. The high sample rate
captures the impulse of breaking waves, and allow them to be studied in
detail. Breaking waves are measured and quantified in the open water. We
measure breaking waves in the surf and recover the trajectory of waves
breaking in the field to a higher degree than previously done. The time series
of surface elevation, and accurate positioning, permits adjacent buoys to be
correlated in a coherent phase-resolved way. Finally, we offer an explanation
and solution for the ubiquitous low-frequency noise in IMU-based buoys and
discuss necessary sampling and design to measure in areas of breaking waves.

##  1 Introduction

Wave breaking is the primary process by which energy, momentum and gases are
exchanged between the air and the sea. It is the limiting factor in wave
growth and helps transfer energy through the wave spectrum from shorter to
longer waves (Melville, 1996; Babanin, 2009; Deike, 2022). The forces induced
by breaking waves are one of the dimensioning factor in marine and nautical
engineering, and their effect on coastlines is profound. Most maritime
activity happens close to the coast, with more than 60 % of humans living near
the coast (Boehm et al., 2017). Estimates further suggest that $80$ % of all
search and rescue cases are found within $20$ km of the shoreline (Breivik and
Allen, 2008). Winds, waves, and currents are heavily modified and influenced
by the coastline where waves and currents interact strongly (Halsne et al.,
2022, 2023). Consequently, the regions with the most complex wave conditions
typically hold the greatest societal importance and are also the most
difficult to sample.

Of the many ways to measure waves, in situ heave buoys and wave staffs are the
most popular (Holthuijsen, 2010). There exists a range of high–quality
commercial moored and drifting wave buoys [see, e.g., the buoys by Datawell
(Datawell BV, 2023), SOFAR Ocean (Raghukumar et al., 2019), and Aanderaa
(Dorgeville et al., 2018), LainePoiss (Alari et al., 2022)]. These buoys are
designed for operational application and therefore have a need to function
robustly for long stretches of times. They usually report statistical
properties, like significant wave height, and in some cases time series up to
a few Hz. Often the time series are only available onboard, and not over
telemetry. However, the last decade has seen a small revolution in low-cost,
open-source wave-measuring devices (Thomson et al., 2023; Rabault et al.,
2020; Feddersen et al., 2023a). These designs generally do not seek to have
the same robustness as commercial buoys, but can measure in quite different
conditions. Because of the low cost and lower environmental impact, they are
also often considered expendable. This is important when measuring in regions
where the risk of loss and damage is high. More importantly, these instruments
are typically an one or more orders of magnitude smaller and lighter than
traditional wave buoys and can thus measure in locations where traditional
buoys cannot. Crucially, they also offer the opportunity to measure quantities
that larger buoys are incapable of capturing, such as the acceleration in
breaking waves (Feddersen et al., 2023b). The limits and capabilities of the
instrument determine which questions can be answered. Heterogeneous and
different solutions are valuable in themselves in that they have different
biases, cover different use cases and make results more robust.

A relevant example to this work is Brown et al. (2018) who previously
quantified breakers in the open ocean by measuring accelerations. Similarly,
the microSWIFT (Thomson, 2012; Thomson et al., 2019) has been used to study
waves near the shore and in areas with strong currents (Rainville, 2022;
Rainville et al., 2023). Sinclair (2014) and later Brown and Paasch (2021) and
Fisher et al. (2021); Feddersen et al. (2023a) have attempted to measure
breaking waves in the surf or in wave laboratories.

In this study we describe and demonstrate a small, lightweight, buoy, dubbed
the Small Friendly Buoy (SFY), tailored to coastal waters and designed for
capturing and quantifying breaking processes as well as spatially varying
wave-current interactions. The code and hardware is open source and available
publicly111https://github.com/gauteh/sfy. The work builds on the OpenMetBuoy
(Rabault et al., 2020).

Some open-source buoys have telemetry so that the data can be transmitted to
shore. This is typically done using Iridium satellites and the Iridium Short
Burst Data (SBD) system, which admit only very low bandwidth (except with
high-power, heavier, equipment). Some buoy designs can only store the
information on board and thus require the buoy to be retrieved. This is often
not possible or requires much time and resources. Loss of data is also an
issue with such an approach, especially for deployments in rough conditions.
The SFY avoids both these problems by relying on the cellular network to
transmit data. This allows us to transmit much more data so that we can send
continuous high-frequency time series of the full three-dimensional
acceleration vector of the buoy. Additionaly, this allows us to build buoys at
a much lower cost and with a smaller physical and environmental footprint.
Together this makes it possible to analyse the full time series for impulses
incurred by breaking waves, and to resolve the phase of waves between buoys
deployed in dense spatial networks. In the coastal environment, resolving this
spatial variability is as important as (if not more than) having a single
high–quality point measurement. The disadvantage is that the deployments must
be within the cellular coverage and will not work far from the coast.

### Measuring the wave field through acceleration at a point

The working principle of the SFY wave buoy is to measure the acceleration in
the wave field, and to convert acceleration as a function of time to
displacement as a function of time. This is an established method that has
been used for many decades (see e.g. Tucker (1958)), initially with mechanical
accelerometers, and more recently with increasingly cheap, accurate and easily
available micro-electromechanical system (MEMS) accelerometers and gyroscopes
on inertial measurement units (IMU). Since acceleration ($a$) is the second
derivative of displacement, we can recover the displacement ($\eta$) by
integrating the acceleration twice,

$\displaystyle v(t)=v_{0}+\int_{0}^{t}a(t^{\prime})\mathrm{d}t^{\prime},$ (1a)
$\displaystyle\eta(t)=z_{0}+\int_{0}^{t}v(t^{\prime})\mathrm{d}t^{\prime}.$
(1b)

In practice, some signal processing is required to get a reliable signal, as
the measured acceleration typically contains low-frequency noise that would be
magnified by the double integration. The common practice is to use some
variant of a high-pass filter to remove lower frequencies than those we expect
to be present in the wave field (lower than around $0.03\textrm{--}0.10$ Hz,
corresponding to wave periods in the range $10\textrm{--}30$ seconds, see,
e.g., Tucker (1958); Waseda et al. (2018)). For this, we detrend the signal
and apply a Butterworth-filter of 10th order before each integration and
perform each integration in the frequency domain (discarding $\omega=0$),

$\mathcal{F}\\{\eta(t)\\}(\omega)=-\omega^{-2}\;\mathcal{F}\\{a(t)\\}.$ (2)

Here, $\omega=2\pi f$ is the angular frequency [rad/s], $f$ is the linear
frequency [Hz] and $\mathcal{F}$ is the Fourier transform. When integrating
spectra of acceleration density ($A$) to elevation energy (variance density
spectrum), $E$, Eq. (2) becomes

$E(\omega)=\omega^{-4}A(\omega),$ (3)

in which case it is not necessary to filter and detrend the signal to analyse
the spectrum. Assuming the sea-surface elevation is a stationary, random
Gaussian process, the characteristics of the variance density spectrum is
defined by its moments $m_{n}=\int_{0}^{\infty}f^{n}E(f)df$. And the
significant wave height in deep-water can be estimated as (see e.g.
(Holthuijsen, 2010) Chapters 3.5 and 4 for a thorough explanation):

$H_{m0}\simeq 4\sqrt{m_{0}}.$ (4)

There are inherent limitations to measuring a single point of the surface,
which cannot capture the changing interference patterns produced by the
dispersive wave field (Magnusson et al., 1999). These are better resolved with
methods that instantaneously span the spatial field, like stereo-video
measurements (see, e.g., Benetazzo (2006); Malila et al. (2022); Malila
(2022)) or densely sampled point measurements (e.g Donelan et al. (1985,
1996)). Additionally, the motion of wave buoys tend to underestimate the wave
crests (Forristall, 2000); moored buoys are affected by the pull of the
anchoring and have a tendency to skirt around the highest wave crests, while
drifting buoys overestimate the mean water level by spending too much time
near the crests.

An inherent assumption of the accelerometer-based approach is that the
acceleration experienced by the buoy is the same as the acceleration of the
water surface. This assumption starts to break down when the wave oscillations
are so rapid that the buoy can no longer reasonably be said to remain at its
equilibrium level of immersion. The frequency at which the buoy no longer
manages to follow the sea-surface is a function of its shape and buoyancy and
can be approximated using the heave response frequency, $f_{\mathrm{hr}}$
(Thomson et al., 2015),

$\displaystyle f_{\mathrm{hr}}$ $\displaystyle=\frac{1}{2\pi}\sqrt{\frac{g\rho
V}{ML}}.$ (5)

Here $\rho$ is the density of seawater, $V$ is the displaced volume, $L$ is
the height of the buoy, and $M$ is the mass of the buoy and the added mass
caused by drag on the water.

The SFY buoy comes in a small free-drifting form and in a larger moored form.
The free-drifting buoy has a heave response of approximately
$f_{\mathrm{hr}}=3.56$ Hz, while the moored buoy has $f_{\mathrm{hr}}=1.44$ Hz
(assuming the buoy is 50 % submerged). Clearly, though, the expression for the
heave response is an approximation, and does not take the horizontal length of
the buoy into account. At a rule-of-thumb level, one can argue that if the
wavelength ($\lambda$) of the wave to be measured is shorter than twice the
(horizontal) radius of the buoy, $r$, the buoy will not be able to follow the
wave. In deep water, the linear phase speed is $c_{\mathrm{p}}=\sqrt{g/k}$
with $k=2\pi/\lambda$ the wavenumber. From this we find that the geometric
cut-off frequency ($f_{\mathrm{gc}}$) where the wavelength equals $2r$ is

$\displaystyle
f_{\mathrm{gc}}=\frac{\sqrt{g/k}}{\lambda}=\sqrt{\frac{g}{2\pi\lambda}}.$ (6)

For the free-drifting buoy $f_{\mathrm{gc}}=2.24$ Hz, and for the moored buoy
$f_{\mathrm{gc}}=1.09$ Hz. These frequencies will be compared with the spectra
from experiments in Sections 3 and 4.

### Structure of this paper

This paper is organised as follows. We first present the design of the SFY
buoy in Section 2. Wave tank experiments of the buoy response are presented in
Section 3. Field experiments from a mooring in open-ocean conditions are
presented in Section 4 together with a comparison against a high-resolution
wave model. Results from surf-zone deployments are presented in Section 5.
Finally, we discuss our findings and compare with other light-weight buoys in
Section 6, and present our conclusions in Section 7.

##  2 Buoy design

The SFY wave buoy consists mainly of three parts; (i) the electronics board
with sensors, processing and telemetry (presented in Section 2); (ii) the
power source, usually alkaline or lithium batteries (see appendix A); (iii)
the housing which protects the other parts and keeps the buoy floating.
Various types of housing can be used, but in this work two types are
described, an 800 ml bottle for free-drifting buoys, and a $60\times 80$ cm
inflatable fender for moored buoys.

### Sensor, processing and telemetry

The circular electronics board measures 60 mm in diameter and about 2 cm in
height. Figure 1 shows the components with the 800 ml bottle used for housing.

The sensor pipeline generally follows the setup for the OpenMetBuoy (Rabault
et al., 2022), hereafter OMB, and benefits from the extensive field testing
and validation that the OMB has been subjected to. The inertial measurement
unit (IMU) used is either the ISM330DHCX or LSM6DSOX (STMicroelectronics,
2019, 2020), both of which yield gyroscope and accelerometer readings. The IMU
low-pass filters the signal internally according to the configured output
rate. The theoretical noise level of the ISM330DHCX is typically $60\,\mu
g/\mathrm{Hz}^{1/2}$ (maximum $100\,\mu g/\mathrm{Hz}^{1/2}$), according to
the manufacturer datasheetSTMicroelectronics (2020). Here,
$g=9.81\,\mathrm{m\,s^{-2}}$ is the earth’s gravitational acceleration.

Accelerometer and gyroscope readings from the IMU are fused by applying an
Attitude and Heading reference system (AHRS) algorithm implemented with a
Kalman filter so that the acceleration is obtained in a North-East-Down (NED)
frame of reference (NXP Semiconductors, 2016; Adafruit, 2023). No magnetometer
is currently included, so the North-East vector points in an arbitrary
direction, but remains stable on short time scales thanks to the measurements
provided by the gyroscope. In this analysis only the relative horizontal
acceleration is utilized. The IMU is sampled at $208$ Hz and the samples are
then fed into the Kalman filter at this same frequency. Finally, the output
from the Kalman filter is filtered using a finite-impulse response (FIR) low-
pass filter (129 taps with a Hamming window) and decimated down to $52$ Hz or
$26$ Hz depending on the application.

It is difficult to quantify the theoretical noise level of the processing
chain as a whole since both the accelerometer and gyroscope introduce noise
before these are fused in a complex nonlinear way by the Kalman filter. Hence,
the noise level of the accelerometer is used as a lower bound of the noise and
as an estimate of its frequency dependence. The noise level of the sensors and
the processing pipeline as a whole is discussed in more detail by Rabault et
al. (2023). Details on how the data are packed, transmitted and received are
described in appendix C.

Figure 1: The components that make up the SFY wave buoy. From the left:
housing (plastic bottle) with foam for additional stability, battery pack
(Alkaline), main board with SD-card, Microcontroller Unit (MCU) in red, modem
and GPS in green, IMU, and GPS and cellular antennas.

##  3 Lab experiment: Wave flume

Experiments were carried out in a wave flume at SINTEF Ocean in Trondheim,
Norway (described in Singsaas et al. (2020)). The flume is 12 m long and 0.53
m wide, with a piston-type wave maker from Edinburgh Designs (Edinburgh
Designs, 2023) at one end, and a perforated absorbing beach at the other end.
The wave maker is programmable and can be set to generate simple harmonic
waves as well as wave packets and random realisations of a specified wave
spectrum.

The flume is filled with filtered seawater from an intake at approximately 80
m depth in the Trondheim fjord. During the experiments described here, the
flume was filled to a depth of 80 cm. Wave gauges can be mounted at any
position in the flume, allowing the surface elevation to be sampled at
frequencies up to 128 Hz. The wave gauges, also produced by Edinburgh Designs,
are based on the principle of measuring resistance between two vertical
parallel metal rods. Table 1 describes the configuration of the five buoys and
wave gauge used.

Instrument |  | Frequency |  | IMU
---|---|---|---|---
| IMU | Output | Real |
Wavebug16 | 52 Hz | 52 Hz | 48.58 Hz | LSM6DSOX
Wavebug23 | 52 Hz | 52 Hz | 48.60 Hz | LSM6DSOX
Wavebug25 | 208 Hz | 52 Hz | 49.07 Hz | ISM330DHCX
Wobbler001 | 52 Hz | 52 Hz | 55.90 Hz | ISM330DHCX
Wobbler002 | 52 Hz | 52 Hz | 55.80 Hz | ISM330DHCX
Wave gauge |  | 128 Hz | 128.00 Hz |
Table 1: Buoy and wave gauge configuration for lab experiment.

(a) $f=0.10$ Hz, $A=3.0$ cm (run 6).

(b) $f=0.15$ Hz, $A=5.0$ cm (run 5).

(c) $f=0.25$ Hz, $A=5.0$ cm (run 4).

(d) $f=0.3$ Hz, $A=5.0$ cm (run 3b).

(e) $f=0.5$ Hz, $A=5.0$ cm (run 2).

(f) $f=1$ Hz, $A=5.0$ cm (run 1).

(g) $f=1.5$ Hz, $A=2.0$ cm (run 8a), _B16_ fixed with rope.

Figure 2: Welch spectrum measured by buoys and wave gauge (WG) measured in a
wave flume with piston configured at different frequencies and wave
amplitudes. The red vertical line marks frequency where the wavelength is
twice the size of the collar width of the buoys ($2.24$ Hz, see Eq. (6)). The
vertical dashed lines mark the input frequency and its higher harmonics.

The Welch spectrum in Figure 2 is calculated with a segment length of 4096 for
the buoys, and 8192 for the wave gauge which samples at approximately twice
the frequency of the wave-buoy.

The measured peaks in the spectra (Figure 2) match the configured piston
frequency and its higher harmonics (gray vertical lines) well. The frequency
of the wave components measured with the wave gauge also match very well. In
the initial experiments the buoys were freely moving at different positions in
the wave flume and were subjected to considerable lateral movement and
rotation due to Stokes drift, so the phase between the buoys should not be
expected to match. Additionally, the vertical amplitude of the generated wave
varied considerably laterally along the flume, probably due to interference
and reflection (particularly for the lower frequencies).

At approximately $2.2$ Hz there is a marked decrease in the wave-buoy spectra
compared to the wave gauge. The collar-width of the buoy (Figure 1) is about
$15.5$ cm. From Eq. (6), we see that a wavelength twice that has a deep-water
frequency of $2.24$ Hz (red vertical line in Figure 2). The measured amplitude
is therefore increasingly dampened as the wavelength approaches the diameter
of the buoy. According to Eq. (5), the heave response frequency is
$f_{\mathrm{hr}}=3.56$ Hz, but it is very sensitive to the degree of immersion
and displacement volume. It seems that the dominating effect on the heave
response is the diameter of the buoy, rather than the weight and volume.
Yurovsky and Dulov (2020) observe a similar geometric cut-off frequency of
about $2.5$ Hz (corresponding to the wavelength twice the hull diameter of
their buoy). As they mention, this means that shorter waves can be measured
compared to most other alternatives (e.g., $1.2$ Hz for the Spotter buoy
(Raghukumar et al., 2019)).

The buoys measure acceleration and have an approximately constant noise level
on the measured acceleration. The wave gauge measures displacement and has an
approximately constant noise level on the displacement. The difference becomes
apparent in the spectrum (Figure 2) at lower frequencies, where the noise
level of the wave gauge is flat (horizontal), while the noise level of the
wave buoys is proportional to $f^{-2}$ following the double integration from
acceleration to displacement. At lower frequencies the wave buoys therefore
have a higher noise ratio compared to the wave gauge, whereas at higher
frequencies the wave buoy performs better since acceleration is better for
measuring high frequency waves (with higher acceleration than displacement),
and conversely the wave gauge is better at measuring low-frequency waves. See
Sec. 1 and 66.1 for more details.

###  3.1 Significant wave height

Piston |  $H_{m0}$ (m)
---|---
$f_{p}$ (Hz) | Run |  $A_{p}$ / $H_{m0}$ (m) | WG | B16 | B23 | B25 | WOB1 | WOB2
0.10 | 6 | 0.03 / 0.08 | 0.06 | 0.01 | 0.07 | 0.01 |  | 0.01
0.15 | 5 | 0.05 / 0.14 | 0.04 | 0.17 | 0.01 | 0.18 | 0.18 | 0.16
0.25 | 4 | 0.05 / 0.14 | 0.11 | 0.08 | 0.08 | 0.09 | 0.08 | 0.10
0.30 | 3a | 0.05 / 0.14 | 0.12 |  | 0.13 |  | 0.09 |
0.30 | 3b | 0.05 / 0.14 | 0.12 | 0.08 |  | 0.08 |  |
0.50 | 2 | 0.05 / 0.14 | 0.10 | 0.12 |  | 0.12 |  |
1.00 | 1 | 0.05 / 0.14 | 0.15 |  | 9.60 |  | 0.15 |
1.50 | 7a | 0.03 / 0.08 | 0.08 | 4.68 | 7.11 | 4.23 |  | 2.14
1.50 | 7b | 0.03 / 0.08 | 0.08 | 3.43 |  |  |  |
1.50 | 8a | 0.02 / 0.06 | 0.08 | 1.78 |  |  |  |
Table 2: Significant wave height measured in the wave flume, automatic high-
pass cut-off frequency.

In Table 2 the significant wave height is calculated from the zeroth moment of
the elevation variance spectrum ($H_{m0}$, Eq. 4) and compared to the
programmed wave amplitude of the piston, and the significant wave height
($H_{m0}$) of the wave gauge (WG). The theoretical $H_{m0}$ for the generated
signal with a single harmonic wave with amplitude $A_{p}$ and variance
$\frac{1}{2}A_{p}^{2}$ is calculated as (see Holthuijsen (2010), Chapter 3.5):

$\displaystyle m_{0}$ $\displaystyle=\int^{\infty}A_{p}\;\delta(f_{p})\;df$
(7a) $\displaystyle H_{m0}$ $\displaystyle=4\cdot\sqrt{\frac{1}{2}A_{p}^{2}}$
(7b)

The measured signal from the buoys need to be high-pass filtered at a cut-off
frequency to avoid low-frequency noise. The wave gauge is not filtered, since
it has a flat noise curve on displacement (Figure 2). Automatic selection of
low-end cut-off frequency is generally hard. This is a problem that affects
IMU based buoys in general (see, e.g., Nose et al. (2018)) and will be
discussed further in Section 66.1. Alari et al. (2022) chose the bandpass cut-
off frequencies of the measured signal in the wave tank tests to be $0.30$ and
$1.28$ Hz and the _LainePoiss ®_ buoy was evaluated at frequencies between
$0.57$ and $0.73$ Hz.

In Table 2 the cut-off frequency is found automatically by searching for the
first spectral minimum above $0.05$ Hz. The algorithm is based on the process
outlined in Rabault et al. (2022) (Figure 7) as well as in Tucker (1958);
Waseda et al. (2018). However, when this algorithm is unsuccessful it results
in unphysically large amplitudes and significant wave height estimates. The
auto-detected cut-off frequency is shown and listed in Figure 2. For instance
for $1$ Hz in Table 2 and Figure 2(f) the significant wave height for _B23_ is
obviously wrong and overestimated compared to _WOB1_. This can be attributed
to the cut-off frequency being chosen too low for _B23_ , while better for
_WOB1_ (see $f_{c}$ in Figure 2(f), note that the cut-off for _B23_ is off the
scale of the plot). In general it is easier to auto-detect the cut-off
frequency when the waves are higher and the signal to noise ratio is greater.
Since we know the input frequency for these experiments we can manually choose
cut-off frequencies. In Table 3 the significant wave height is shown for the
manually chosen frequency band. These values match better, and do not show
exaggerated significant wave heights like those present in Table 2 (where a
too low or too high cut-off frequency has been picked).

Piston | Bandpass |  $H_{m0}$ (m)
---|---|---
$f_{p}$ (Hz) | Run |  $A_{p}$ / $H_{m0}$ (m) | $f_{hp}$ | $f_{lp}$ | WG | B16 | B23 | B25 | WOB1 | WOB2
0.10 | 6 | 0.03 / 0.08 | 0.05 | 1.28 | 0.06 | 0.03 | 0.07 | 0.03 |  | 0.03
0.15 | 5 | 0.05 / 0.14 | 0.05 | 1.28 | 0.04 | 0.17 | 0.07 | 0.18 | 0.18 | 0.16
0.25 | 4 | 0.05 / 0.14 | 0.10 | 1.50 | 0.11 | 0.08 | 0.08 | 0.09 | 0.08 | 0.12
0.30 | 3a | 0.05 / 0.14 | 0.10 | 1.50 | 0.12 |  | 0.13 |  | 0.09 |
0.30 | 3b | 0.05 / 0.14 | 0.10 | 1.50 | 0.12 | 0.08 |  | 0.08 |  |
0.50 | 2 | 0.05 / 0.14 | 0.30 | 2.00 | 0.10 |  | 0.12 |  | 0.12 |
1.00 | 1 | 0.05 / 0.14 | 0.30 | 2.00 | 0.15 |  | 0.43 |  | 0.15 |
1.50 | 7a | 0.03 / 0.08 | 0.30 | 2.00 | 0.08 | 0.18 | 0.23 | 0.15 |  | 0.08
1.50 | 7b | 0.03 / 0.08 | 0.30 | 2.00 | 0.08 | 0.11 |  |  |  |
1.50 | 8a | 0.02 / 0.06 | 0.30 | 2.00 | 0.08 | 0.10 |  |  |  |
Table 3: Significant wave height measured in the wave flume, custom bandpass
cut-off frequencies.

###  3.2 Wave buoy orbital measurements

To determine the buoy orientation along the wave-flume we assume that the
buoys movement is mainly varying along the wave direction. Since we do not
have a magnetometer we do not expect the $x$ and $y$ axes to be pointing in
the same direction on long time-scales (limited by gyroscope drift). Figure 3
shows the vertical motion as a function of time for a sine wave of $0.3$ Hz
(Figure 2(d)), as well as the vertical and horizontal acceleration.

In order to obtain the trajectory of the wave buoy we apply principal
component analysis (PCA) to the high-pass filtered $x$ and $y$ acceleration.
The acceleration is filtered to remove drift and only consider the movement
due to wave motion. The first principal component, $\bf u_{0}$, is the
direction in the $xy$-plane along which the variation is greatest, the second
component ($\bf u_{1}$) is orthogonal to $\bf u_{0}$ representing the rest of
the variance.

We project the $x$ and $y$-accelerations onto the normalized first principal
component ($\frac{\bf u_{0}}{\left\|\bf u_{0}\right\|}$, horizontal) and the
normalized second principal component ($\frac{\bf u_{1}}{\left\|\bf
u_{1}\right\|}$, vertical), the original movement in the $xy$-plane and the
re-projected movement is shown in Figure 4:

$\begin{bmatrix}|\\\ \bf h_{i}\\\ |\end{bmatrix}=\begin{bmatrix}|&|\\\ \bf
x&\bf y\\\ |&|\end{bmatrix}\cdot\frac{\bf u_{i}}{\left\|\bf u_{i}\right\|}$
(8)

$\bf h_{0}$ and $\bf h_{1}$ are the vectors with re-projected $x$ and $y$
accelerations.

If the buoy is only moving along one direction, and the gyroscope is not
drifting, all movement should be aligned along the first principal component.
It is clearly visible in Figure 4 that this is the case for almost all the
movement. The ratio between the two principal components for 10 seconds in the
wave-flume is $925$ (upper panel in Figure 4), which provides a measure of how
robust this method is on this time scale.

Figure 3: Vertical motion, vertical acceleration and horizontal acceleration
along direction of maximum variance for a $0.3$ Hz sine wave. _Upper:_
Vertical displacement for the wave-gauge, B23 and B25. _Middle:_ Vertical
acceleration for B23 and B25. _Lower:_ Horizontal acceleration for B23 and
B25. Figure 4: Trajectory of the _B25_ wave-buoy in the wave-flume with 10
seconds of $0.3$ Hz sinusoidal waves (Figure 2(d)). _Upper:_ _xy_ -plane with
principal components and re-projected trajectory. _Middle:_ high-pass filtered
trajectory in horizontal–vertical plane along first ($\bf u_{0}$) and second
($\bf u_{1}$) principal component. _Lower:_ full bandwidth trajectory in
horizontal–vertical plane for first principal component.

Linear waves predict the orbital motion of particles. If the wave-buoy is to
measure the details of wave-motion it should be water-following. The relation
between the horizontal and vertical motion, and their spectra, will tell us
how well the buoy is following the particle motion in a wave (Feddersen et
al., 2023a). The amplitudes of the horizontal and vertical particle orbital
velocity in a plane wave are (e.g. Holthuijsen (2010))

$\displaystyle u_{x}$ $\displaystyle=\omega a\frac{\cosh[k(d+z)]}{\sinh(kd)}$
(9) $\displaystyle u_{z}$ $\displaystyle=\omega
a\frac{\sinh[k(d+z)]}{\sinh(kd)}.$ (10)

Here, $d$ the local water depth and $k$ the wave number. The ratio of
horizontal to vertical velocity is thus

$\frac{u_{z}}{u_{x}}=\frac{\sinh[k(d+z)]}{\cosh[k(d+z)]}=\tanh[k(d+z)].$ (11)

At the surface, $z=0$, Eq. (11) becomes $\tanh(kd)$. The same relation (11)
holds true for the acceleration, $\frac{a_{z}}{a_{x}}$ (differentiate Eq. (9)
and Eq. (10) twice). In this setup $k=0.71\;\textrm{m}^{-1}$ for $f=0.3$ Hz,
and $\tanh(kd)$ is $0.51$, while the orbital motions in Figure 4 (middle
panel) have $\frac{u_{z}}{u_{x}}$ of approximately $0.36$.

###  3.3 Breaking waves

Figure 5: A breaking wave measured in the wave-flume by the buoys drifting
freely next to the wave gauge. The integrated vertical displacement for the
wave gauge (WG), and buoys 23 and 25 are shown. The vertical and horizontal
acceleration (along main principal component, as in Sec. 3.2). Buoy 23 samples
the IMU at $52$ Hz, while buoy 25 samples the IMU at $208$ Hz.

Two observations of the relation between the measured harmonic sine waves and
the breaking waves are of particular interest. First, the maximum horizontal
acceleration for the sine wave is of the order of $\pm 0.05g$ (Figure 3),
whereas for the breaking wave _in the lab_ it exceeds $\pm 1g$. Second, buoys
23 and 25 show similar accelerations and vertical displacements for the non-
breaking wave (upper panel in Figure 3). However, for a breaking wave the
integrated vertical displacement (upper panel in Figure 4) is dominated by
exaggerated low-frequency signals for buoy 23. The major difference between
the two buoys during the recording of the breaking wave is that buoy 23 is
sampling the IMU at $52$ Hz, while buoy 25 is sampling at the full $208$ Hz
before filtering and down-samplng to $52$ Hz after feeding and re-projecting
the acceleration through the AHRS fusion algorithm. These results will be
discussed further in Sec. 66.1.

##  4 Open water: Fedjeosen

A wave buoy built into a fender was deployed at $60.752145^{\circ}$ N,
$4.693160^{\circ}$ E, moored at $90$ m depth, west of Hellisøy lighthouse in
Fedjeosen in July 2023. The location is exposed to to westerly waves from the
North Sea, but is sheltered from the north by Fedje Island. The configuration
is described below (shown in Figures 6 and 19) and is very similar to
recommended moorings described by Martini et al. (2021).

Fedjeosen is the main shipping lane into Hjeltefjorden and the coastal
installations around Bergen. The channel creates a strong tidal current with
spatially and temporally varying current gradients across the channel (which
is about $2$ km wide). This will give rise to strong wave-current interaction
and a spatially inhomogeneous wave field with open-ocean amplitudes. It is
thus a challenging location to test the equipment, but also one where both
storm waves and low-frequency swell can be found, allowing us to test the full
measurement range of the buoy.

The buoy is equipped with $3\times 21=63$ D-cell Alkaline batteries,
sufficient for a one-year deployment with transmissions over the cellular
network of measurements sampled at $26$ Hz.

(a) Location of mooring in Fedjeosen, map from Kartverket (sjøkart).

(b) A fender of dimensions $65$ cm x $88$ cm floats at the surface, and houses
the electronics and batteries.

Figure 6: Fedjeosen mooring.

The buoy was deployed “anchor last” with an anchor of approximately $80$ kg.
As significant wave heights of more than $12$ m can be expected during winter
storms, more weight is required than in sheltered locations. Still, the weight
is almost two orders of magnitude less than for large wave buoys, making the
deployment manageable with a smaller vessel. The mooring rope is $12$ mm
Danline Polypropylene (PP). This diameter dimension is excessive, and causes
unnecessary drag from currents, and a smaller diameter should be chosen in the
future. The buoy is deployed by first lowering the floating buoy to the water
surface, then the mooring is unspooled while traveling against the weather.
Finally the anchor can be lifted off the ship and be dropped with a releasable
hook. The rope is weighted with a steel wire of approximately $10$ kg between
the surface buoy and a buoyancy float at $30$ m so that the buoy is relatively
free to move with the waves, and the rope is kept out of the way for vessels
traveling around the buoy. The bottom anchor is split in two parts with some
chain as buffer, so that in large waves the first chain and anchor will move
first in order to prevent the entire anchor from being lifted and wandering
off position (see schematic in appendix D). The cellular modem connects to a
station $80$ km away (more details in appendix B).

Figure 7: $H_{m0}$ measured by a moored buoy at the western coast of Norway
(Fedjeosen), compared to operational wave models (with and without current)
(The Wamdi Group, 1988), satellite observations of significant wave height,
and the measured wind nearby. Figure 8: Observations from the buoy compared to
the simulated $H_{m0}$ (WAM800 with current). Linear regression and quantile-
quantile (Q-Q) lines are shown. The model predict slightly higher significant
wave height, but with a otherwise fairly linear relationship (Pearson
correlation coefficient, $R=0.94$). The model without current shows the same
$R$-value.

The measured significant wave height is compared to operational coastal wave
models (Fig. 7 and 8), with and without currents (The Wamdi Group, 1988). The
nearest grid point in the wave model is about $280$ m away from the buoy
location. Figure 8 compares the observations against the simulated $H_{m0}$:
there is a good match with an $R$-value of $0.94$ for both models. The buoy
measures slightly lower $H_{m0}$ values than model predicts, but the
relationship is close to linear.

Deploying the SFY at an exposed location like Fedjeosen provides the
opportunity to compare its recordings with satellite observations of
significant wave height to add an additional reference for the comparison.
Measured $H_{m0}$ was compared to the near real time L3 satellite altimeter
measurements from the Copernicus Marine Service catalogue
product222WAVE_GLO_PHY_SWH_L3_NRT_014_001 (E.U. Copernicus Marine Service
Information . Marine Data Store (MDS)(2023), CMEMS). The satellite data are
retrieved and collocated with the SFY using the open source tool
wavy333https://wavyopen.readthedocs.org following the collocation routine with
temporal and spatial constraints as outlined by Bohlinger et al. (2019). To
acquire enough data, we set the temporal constraints to $\pm 10$ minutes of
the time stamp of the significant wave height, and a maximum spatial distance
of $100$ km. This resulted in matches with 6 different satellites, namely
Sentinel-3A/B, Cryosat-2, CFOSAT, SARAL-AltiKa, and Sentinel-6A Michael
Freilich. Satellite observations appear to follow closely the time series of
$H_{m0}$ recorded by the SFY (Fig.7).

(a) Hjeltefjorden ($52$ Hz), logarithmic axes.

(b) Fedjeosen ($208$ Hz), logarithmic axes.

(c) Hjeltefjorden ($52$ Hz), linear axes.

(d) Fedjeosen ($208$ Hz), linear axes.

Figure 9: Elevation energy (variance density) spectra for comparable
situations (similar $H_{m0}$ and $T_{p}$) of the same buoy (Figure 6(b))
deployed at two different locations. The IMU (before feeding the AHRS-
algorithm) was configured to sample at $52$ Hz in the first case
(Hjeltefjorden), and at $208$ Hz in the second case (Fedjeosen). The arrow in
the lower left panel shows low-frequency noise caused by insufficient sample
rate.

In Figure 9 spectra for 20-minute windows of the time series are compared to
spectra for the same buoy deployed inside Hjeltefjorden a few months earlier.
The deployment was very similar except that for the first deployment the IMU
was sampling at $52$ Hz, while for the second (in Fedjeosen) the IMU was
sampling at $208$ Hz and downsampled after being passed through the AHRS-
algorithm. The spectra (Figure 9) have been chosen for similar wave situation
in terms of $H_{m0}$ and $T_{p}$. A notable difference is that for the buoy
configured to sample at the lower frequency ($52$ Hz), more low-frequency
noise is observed (arrow in Figure 9(a)). This difference will be discussed
more in Sec. 66.1.

Between $1$ and $1.1$ Hz, after the $f^{-4}$ slope there is a pole (onset of
increased negative logarithmic slope). This corresponds with the frequency of
wavelength twice the horizontal diameter of the buoy ($65$ cm) which is $1.09$
Hz. It therefore seems that the dominant factor capping the upper measurable
frequency is the size of the buoy.

### Breaking waves measured by the moored buoy

Breaking waves hitting or hurling a buoy cause rapid acceleration impulses as
can be clearly seen in the wave flume (Figure 5). The strength of this impulse
depends on buoy dimensions and wave steepness and how the breaking wave
impacts the buoy. Seldal (2023) did a preliminary investigation into the
signature of the breaking waves to try distinguish the different types of
breakers in a way similar to the approach of Brown and Paasch (2021). The
simplest method of quantifying the number of breakers would be to count maxima
in the time series of horizontal acceleration, but this would detect many
peaks for each breaking event. In order to make the method more robust against
false positives we apply the method from Brown et al. (2018). Here we
calculate the Short-Time Fourier-Transform (STFT) for tapered windows of
$t_{\textrm{bw}}=1.5$ s (overlap between windows of $t_{bw}/2$) for all three
components of acceleration ($a_{z}$, $a_{x}$, and $a_{y}$):

$A(f,t^{\prime})=|\mathcal{F}\\{a_{z}\\}|+|\mathcal{F}\\{a_{x}\\}|+|\mathcal{F}\\{a_{y}\\}|\quad\textrm{for
time-windows}\;\left<t^{\prime},t^{\prime}+t_{bw}\right>$ (12)

and take the sum ($\alpha$) of the absolute magnitude along the frequency
dimension and the sum ($A$) of components for frequencies above
$f_{\mathrm{bc}}=3.6$ Hz:

$\alpha(t)=\sum\limits_{f>3.6}A(f,t).$ (13)

Regular, linear waves, do not exist at significant energy levels at
frequencies greater than $2$ Hz and we here assume that the energy found at
these frequencies is caused by impulsive accelerations from breaking waves.
The frequency was chosen so that it is above the consistent band of energy
(upper panel in Fig. 10(a)).

(a) Detected breaking events for the moored buoy in Fedjeosen. _Upper:_ The
sum of the acceleration spectrum ($A$). _Middle:_ Detected breakers and
$\alpha$. _Bottom:_ Breaker fraction ($Q_{b}$) for 20 minute windows.

(b) 2023-10-14T16:00

(c) 2023-10-24T12:00

(d) 2023-11-04T12:15

Figure 10: Breaking events and photos taken towards the buoy position (located
to the left behind the hill) at times marked in Figure 10(a). Photos by Norsk
Luftambulanse, used with permission.

Breakers are identified by counting peaks of $\alpha$ above a threshold value.
The parameters chosen for window length ($t_{\mathrm{bw}}$), $f_{\mathrm{bc}}$
and the threshold for $\alpha$ are dependent on the buoys response to breakers
and signal processing (Brown et al., 2018). As in Brown et al. (2018) we use a
threshold $\alpha=3.25\,\mathrm{m}\,\mathrm{s}^{-2}$, and rather increase the
cut-off frequency to $3.6$ Hz with the argument that only counting high-
frequency impulses is a more robust way of detecting the breakers. Along with
the plot of $\alpha$ and the detected breakers in the middle panel in Fig.
10(a), we also plot the norm of the acceleration components
$\sqrt{a_{z}^{2}+a_{x}^{2}+a_{y}^{2}}$, filtered above $2$ Hz. It gives very
similar results to $\alpha$, with the main difference caused by the smoothing
due to the windowing in $\alpha$ (middle panel in Fig. 11).

In Figure 11 broadbanded maxima in acceleration (visible as vertical stripes
in the acceleration spectrum) are used to detect breakers. In the lower panel
the three acceleration components are shown (unfiltered). Similar impulses to
the lab experiments are apparent (Fig. 5).

Figure 11: Detected breaking events for the moored buoy in Fedjeosen. _Upper:_
The sum of the acceleration spectrum ($A$). _Middle:_ Detected breakers and
$\alpha$. _Bottom:_ Unfiltered acceleration components.

A way to quantify breaking waves is to estimate the breaking frequency (or
approximately breaking probability) (Schwendeman and Thomson, 2014; Brown et
al., 2018),

$Q_{b}=N_{b}T/\tau.$ (14)

Here, $N_{b}$ is the number of breakers detected during a measurement
interval, $\tau=20$ min, and $T$ is the wave period, here approximated with
$T_{m01}$. Figure 10(a) shows $Q_{b}$ (lower panel) for a subset of the data.
We have visually inspected images taken every 15 minutes with an air traffic
camera on the Fedje Island pointing in the direction of the wave buoy. In
Figure 10 photos from three instants are shown. These are marked in Figure
10(a) with vertical dashed lines. Qualitatively, the detection of breakers
agrees well with the visual impression of breaking waves in the photographs.
The wave steepness ($H_{m0}/(g\cdot\frac{T_{m01}^{2}}{2\pi})$) for the
measurement period varies very little (but agree well with models).

##  5 Arrays of wave buoys in the surf

The buoy was tested in the surf in two separate locations for two types of
experiments. First to record the waves and breaking events simultaneously on
three buoys (Jæren, Norway), and demonstrate that it is possible to study the
phase difference of the events between the buoys in the array. Secondly, to
try to measure the trajectory of a single buoy as it is caught in a breaking
wave (Green Bowl, Bali, Indonesia).

###  5.1 Array of buoys in the surf

The first three buoys were deployed as a free-drifting array on the beach
outside Jæren on the west coast of Norway on the 8th of January 2023. The
cross-shore wind and rip currents carried the buoys outwards and they needed
to be picked up before drifting too far out at sea. Figure 12 shows the drift
trajectory of one of the deployments and the conditions of the day. As the
buoys move independently of each other, their relative positions are
continuously changing (white dashed triangles).

(a) Drift trajectory with the instantaneous geometry shown as dashed white
lines. Aerial images © norgeibilder.no, Kartverket (Rogaland 2019).

(b) Wave conditions on the day. The buoys are visible on the crest, to the
left on the wave that is breaking.

Figure 12: Free-drifting array of buoys in the surf outside Jæren. Figure 13:
Vertical acceleration spectrograms for three buoys, captured in the surf at
Jæren. Vertical lines of high intensity signals spanning the full frequency
spectrum is visible at each buoy, with some delay in phase.

Figure 13 shows the acceleration density spectrogram for the vertical
component of the three buoys for drift shown in Figure 12. Vertical lines of
high energy impulses assumed to be caused by breaking events are visible, and
are correlated in time between the buoys. Since the buoys in this experiment
were configured with very sensitive accelerometer range ($\pm 2g$) and
gyroscope range ($\pm 125$ dps), the sensor saturates and the trajectories are
difficult to recover.

###  5.2 Trajectory in a breaking wave

In a similar experimental setup, a single SFY buoy was deployed in the surf at
Green Bowl, Bali, Indonesia, in November 2023. In this deployment the
acceleration range is set to $\pm 16g$ and the gyroscope range is set to $\pm
1000$ dps, and the AHRS filter is run at $208$ Hz. First the spectrum (Figure
14) was measured for 20 minutes outside the breaking waves as a reference. The
swell was about 0.9 m while the surfing wave height was about 0.3–0.5 m. The
spectrum does not consist of a single peak, so irregular sets of waves
breaking onto the shore is expected.

Figure 14: Spectrum captured outside the surf at Green Bowl, Bali.

The buoy was then deployed so that it would drift towards the shore through
the breaking waves. The horizontal components are re-projected along the
direction of maximum variance as in Sec. 33.2 and integrated to the horizontal
and vertical displacement. A cut-off frequency of $0.05$ Hz is used when
filtering before integrating.

In Figure 16 a wave approaches the buoy (painted red). The buoy is drawn
offshore towards the surf wave _(a)_ , and almost escapes the breaking crest
_(b)_ , but eventually the buoy is caught in the overturning wave _(c)_ and
thrown towards the surface of the water (exceeding $10g$ upwards acceleration
on impact) and flushed out. A second orbit begins as the next (non-breaking)
wave travels towards the shore.

Figure 15: Acceleration and spectrogram at Green Bowl, Bali of a buoy in the
breaking waves of the surf.

(a) 22 seconds

(b) 23 seconds

(c) 24 seconds

(d) Orbital movement of the buoy for the time between vertical lines in Figure
15. The black circle marks the start of the buoy trajectory and each
subsequent second is marked by a red circle. The black triangle marks the end.

Figure 16: Orbital movement of buoy and photos captured at the same breaking
wave recorded by the buoy in Figure 15.

The recorded displacement amplitudes match well with expected values from the
spectrum and visual observations. Usually, measurements from breaking waves in
the surf are not suitable to be integrated to displacement without heavy
filtering. In Figure 15 we can match the photographs with the integrated
trace, and see that the recorded trajectory is sensible even with a very low
cut-off frequency. This is discussed further in Sec. 66.2.

##  6 Discussion

###  6.1 Low-frequency noise and breaking waves

All IMU-based wave buoys will struggle to measure low-frequency signals. The
sensor measures acceleration, and the double integration to obtain vertical
displacement ((1a) and (1b)) introduces drift which must be removed using
detrending. Additionally, a core assumption of AHRS-algorithms is that they
measure a high-frequency signal, and very long period waves may not fall into
this domain and appear to shift the earth’s gravity vector from the buoys
point of view.

The lower frequency signal and noise of the acceleration are amplified
compared to the higher frequencies through the integration, Eqs. (2) and (3).
Displacement measurements from IMUs are therefore inherently sensitive to low-
frequency noise. The low-frequency noise can be seen in Figure 2 and 9, and is
similar to spectra measured by other IMU-based buoys (Nose et al., 2023;
Chuang et al., 2013; Arraigada and Partl, 2006; Collins et al., 2014; Brown et
al., 2018; Veras Guimarães et al., 2018; Rabault et al., 2020; Rainville,
2022). The longest wave periods that can be measured robustly using IMUs are
about $20$ seconds ($0.05$ Hz), depending on the conditions (Brown et al.,
2018; Van Essen et al., 2018; Datawell BV, 2023). Note that up to $100$ s
periods can be observed by using e.g. GPS to measure wave orbital velocities,
although this is also not trivial (Thomson et al., 2018). Note that when
comparing spectra the length of the signal should be similar in order to
expect similar signal to noise ratio.

Consequently, the acceleration must be high-pass filtered with a chosen cut-
off frequency. Choosing this cut-off frequency automatically is not trivial,
since the assumed low-frequency noise often extends higher than $0.05$ Hz and
can be difficult to distinguish from the signal. In this analysis we use a
relatively simple algorithm where we attempt to find a minimum in the smoothed
spectrum after $0.05$ Hz (Nose et al., 2018; Rabault et al., 2022) which is
used as the cut-off frequency (see Sec. 33.1).

In particular, in experiments in the surf-zone we notice that there is often
no significant energy minimum between the main peak ($f_{p}$) and the low-
frequency noise. The same has been observed for sporadic 20-minute periods in
the SFY and OpenMetBuoy when deployed in the open ocean (also observed in
other buoys (Nose et al., 2018)). These spectra tend to give exaggerated
estimates for $H_{m0}$ or invalid estimates for $T_{p}$ (Alari et al., 2022).

Most other assessments of the performance and behavior of IMU based buoys
observe and identify the low-frequency noise (see references above), but its
origin is not well explained. Some of it is attributed to the integrated noise
of the IMU. However, that does not explain why there is much higher low-
frequency noise in some situations than others, or that the noise is well
above the expected noise-floor (when extrapolating the noise curve from high
to low-frequencies). Alari et al. concluded that the low-frequency energy is
misplaced from other frequencies, and that it should not be filtered
completely away for calculation of spectral moments as is the usual way to
deal with this noise. Brown et al. (2018) also attribute some of the low-
frequency noise as alias from higher frequencies. For the MOTUS buoy
(Dorgeville et al., 2018), a mechanical filter was used to reduce measured
vibrations (from wind, waves, and currents) above the Nyquist-frequency that
fold back into lower frequencies. However, for the SFY both the gyroscope and
accelerometer have built-in low-pass filters that would _avoid sampling
alias_.

Another concern is that rotation causes centrifugal acceleration. Feddersen et
al. (2023a) found that a buoy caught in a overturning breaking wave, with the
IMU located fairly close to the center, only undergoes small centrifugal
accelerations. They find that angular velocity of a tumbling buoy very seldom
exceed $600$ dps during breaking events, and never more than $1000$ dps. The
SFY buoy has sampled at ranges of $\pm 125$ and $\pm 500$ dps in the open-
water, and $\pm 1000$ dps in the surf zone (a lower range gives a better
resolution, unless it saturates).

We observe that the low-frequency noise occurs more frequently in the surf,
and only sporadically in the open ocean. As wave breaking is more violent and
happens much more often in the surf zone, it seems likely that high-frequency
acceleration impulses from breaking waves are the cause of the low-frequency
noise in the measured elevation. Specifically, rapid acceleration and rotation
caused by a breaking wave insufficiently sampled into the fusion algorithm
would cause the buoy to lose orientation and would no longer correctly extract
the vertical acceleration. To test this hypothesis we configured buoys with
different IMU sample rates ($208$ Hz and $52$ Hz), but equal output rate
(filtered using a FIR-filter and decimated). In the wave flume (Sec. 3), both
the low-frequency and the high-frequency buoys performed well on harmonic
waves (Figure 3). However, when the buoys are subjected to breaking waves in
the wave flume (Sec. 33.3), the buoy with the higher IMU-sampling (B25)
performs far better than the other (B23), with low-frequency oscillations
clearly visible.

A buoy that encounters breaking waves only sporadically yield a spectrum
contaminated by these impulses. If this is the case, higher sample rate should
better resolve the rotation and acceleration and thus result in less low-
frequency noise. The spectra in Fig. 9 suggest that the buoy with $208$ Hz
sampling and AHRS frequency performs better on low-frequencies than the one
with $52$ Hz sample rate. Detecting and removing segments with breaking
impulses from the time series should also improve the spectral estimates.

Additionally, a buoy in breaking waves requires much higher sample rates and
ranges in measuring rotation and acceleration. Given the large accelerations
measured in breaking waves (several $g$ in the open water, and more than $10g$
in the surf in our case (Sec. 5), and $8--14$ reported by Sinclair (2014),
Brown et al. (2018), and Feddersen et al. (2023a)), a much higher sample rate
is probably required to have a chance at measuring these signals with any
success.

In Sec. 5, Fig. 16(d) we were able to retrieve the elevation with a cut-off
frequency as low as $0.05$ Hz.

###  6.2 Capturing and quantifying breaking waves

A breaking wave is a highly impulsive event. Impulses have infinite bandwidth,
and it is impossible to fully capture them with a band-limited method. Our
experiments in the wave flume (Sec. 33.3), the open water (Sec. 44) and in the
surf (Sec. 5) show that the horizontal and vertical acceleration impulse of a
breaking wave far exceed those of regular waves. The acceleration experienced
by the buoy during a breaking event exceed $10g$, consistent with previous
work (Sinclair, 2014; Brown and Paasch, 2021; Feddersen et al., 2023a). An
important observation is that the high-acceleration impulses are very short
and they will not be adequately captured unless sampled frequently enough.

In the open water (Sec. 4) the breaking fraction is quantified using methods
developed by Brown et al. (2018). The fraction qualitatively agrees with
photos from the location. However, a more controlled experiment with camera on
the buoy, or the buoy located in the field of vision of stereo-video cameras,
would provide more reliable context. This would allow better evaluation of how
robust this method is and the influence of the parameters that go into the it.

Feddersen et al. (2023a) captures the trajectory of a breaking wave generated
at the _Surf Ranch_ remarkably well. In this work we attempt to capture the
trajectory in the waves breaking onto a beach (Sec. 5). The waves breaking on
the beach do not consist of a single reproducible, relatively clean, wave.
Yet, we do capture the trajectory well compared to the spectrum outside the
surf and the photographs. The signal was filtered with a cut-off frequency as
low as $0.05$ Hz, without resulting in exaggerated displacements. To our
knowledge a breaking wave in the ocean has not been captured with a buoy in
this way before.

The reason that the buoy can measure this way is presumably due to a
combination of using a small wave buoy, high frequency sampling, and a wide
measure range on the IMU (accelerometer and gyroscope).

###  6.3 Cellular network and power source

The cellular network has performed remarkably well even when the antennas have
only been a few cm above the surface. A better container with a better aligned
and placed antenna is likely to perform better. In the moored buoy, where the
antenna can be placed higher ($30\textrm{--}40$ cm), the modem connects with a
cellular tower $80$ km away (Fig. 18). Only during the largest waves
($H_{m0}>7$ m) do we see intermittent failures in transmitting data. This will
cause loss of data when the buoy does not have a large enough internal buffer.
A larger buoy might be more stable in this situation and perform better. It is
likely that also satellite-based telemetry will struggle to transmit data in
such sea states.

##  7 Conclusions

The SFY, a small, light-weight, drifting or moored wave buoy design with the
capability to detect breaking events has been described. The buoy is designed
to work in coastal environments and communicates over the cellular network as
well as stores the data on SD-card. Its design, code, and hardware is open
source and available publicly444https://github.com/gauteh/sfy. The buoy has
been proven to work reliably up to $80$ km from the nearest cellular tower,
and in its small form measures frequencies from $0.05$ Hz to about $2.2$ Hz
(or $1.1$ Hz for the moored buoy, without correction). The main advantages of
this buoy are:

  * •

_light-weight_ ($\approx 0.5$ kg) and _low-cost_ ($\approx 200$ USD): allowing
many buoys to be deployed, easy deployment from small vessels or even by
swimmers or surfers.

  * •

_high-bandwidth_ : with a greater bandwidth for data transmission, the full
time series of AHRS-processed high-frequency accelerations ($52$ Hz) is
transmitted back. This allows a much more detailed on-land analysis compared
to transmitting only the wave spectra.

  * •

_open source_ : anyone can build and modify the buoy to adapt it to their
experiment.

Through wave flume experiments, deployments in the open-water, and in the
surf, the buoy is shown to measure significant wave height and other spectral
properties accurately. It is shown to measure the surface elevation in such
detail that the individual wave phases can be compared between adjacent buoys.
These properties enable the buoy to be used in array configurations where the
spatial details of the wave field should be studied, e.g., within a stereo-
video footprint (Malila, 2022) or in areas with strong wave-current
interactions.

The most challenging measurements were performed in the surf zone. The
irregular waves breaking on the beach were not produced by a singular wave
component or soliton. Yet, by measuring at very high frequency ($208$ Hz) and
with a large dynamic range ($\pm 16g$ and $\pm 1000$ dps), we were able to
reconstruct the trajectories of the buoy through breaking and non-breaking
waves on the beach with a higher degree of success than previously done.
Necessary to this is that we were able to use a low cut-off frequency of
$0.05$ Hz here, normally reserved for open-waters. The lack of exaggerated
elevation amplitudes after integrating to displacement gives us confidence
that the buoy measures correctly. In Table 4 we recommend parameter ranges
depending on the deployment scenario.

Scenario | Frequency | Acceleration | Gyroscope
---|---|---|---
Sea-ice |  $100$ Hz | $\pm 2g$ |  $\pm 125$ dps
Open-water |  $208$ Hz | $\pm 4g$ |  $\pm 500$ dps
Surf |  $>208$ Hz | $>\pm 16g$ |  $\pm 1000$ dps
Table 4: Recommended sample rate and IMU range for different scenarios (see
also Sinclair (2014); Rabault et al. (2020); Feddersen et al. (2023a)).

An important conclusion which we surmise will be valid for all IMU-based buoys
is that low-frequency noise can to a large degree be explained by impulsive
events, like breakers, that are not adequately sampled. The inadequate
sampling causes the AHRS algorithm to become confused, and the large, but very
short, acceleration peaks are not well captured. We find that by sampling the
IMU and running the AHRS-algorithm at a higher frequency before downsampling
its output (if bandwidth is limited), much better noise characteristics are
achieved. By this logic, better noise characteristics or estimates based on
the spectrum (like $H_{m0}$) may improve if impulsive events can be detected
and excluded from the spectrum.

The advent of small, disposable drifting wave buoys with telemetry opens new
opportunities for dense sampling of the surf zone and regions with strong
spatial variability, such as tidal currents in open-ocean conditions (see,
e.g., the studies of the wave modulation by the Moskenes tidal current in
Northern Norway by Saetra et al. 2021; Halsne et al. 2022, 2023). Deploying
large, densely spaced, arrays of buoys that sample at rates high enough to
identify breakers and the accelerations in breaking events should allow for
experiments that are currently not possible, or prohibitively impractical and
expensive to do.

###### Acknowledgements.

Jim Thomson, Alex de Klerk, E.J. Rainville and Falk Feddersen are thanked for
discussions and generously sharing their knowledge and experience. Roger
Storvik helped transferring the electronics from schematics to the circuit
board. The following are thanked for their help in deploying buoys in the surf
and technical assistance: Silje Røe, Susanne Moen Olsen, Judith Thu Ølberg,
Daniel Benedet, Max Deffenti. Jan Bjarte Valsvik and the students of Jæren FHS
are thanked for their help in the early experiments. Ragnhild and Ingrid
helped out testing the buoyancy and durability of the early versions of the
buoy. This research and data collection has been supported by the Research
Council of Norway grants _ENTIRE_ (no. 324227) and _Havvarsel_ (no. 310515),
as well as by the Norwegian Coastal Authority. P.B. gratefully acknowledges
funding through the project _COSY – Coastal Operational and Sustainable use of
satellite altimetrY_ , funded by the Norwegian Space Center under grant
74CO2222. The embedded code, bill of materials, tutorials for building the
buoys, hardware schematics and circuit board files are available online at:
https://github.com/gauteh/sfy. The processing code and server side code is
available at the same place. The data from Sec. 4 is available in CF-compliant
NetCDF at:
https://thredds.met.no/thredds/catalog/obs/fouombuoys/fedjeosen_nv/catalog.html.
The data from Sec. 3 and 5 is available at [will be available upon
publication]. [A] Power usage and cellular network range

##  Appendix A Power usage

The majority of the power is drawn by the cellular modem during connection and
transmission of data. When the modem is connecting, current spikes up to $2$ A
can occur for a few milliseconds at a time. Super-capacitors limit this to
about $0.3$ A, and thus enable batteries with higher internal resistance to be
used. Figure 17 shows the measured current (at $4.2$ V) during cellular
connection, transmission, and lower-current logging for a buoy configured to
log at $26$ Hz and with storage on the secure digital (SD) card temporarily
disabled (as in Section 4). The average power consumption for this
configuration is about $32$ mW with an average current going as low as $0.6$
mA when logging to memory only. Note that spikes at the beginning of each
section are artefacts caused by measurement range change in measuring the
current.

Considerable effort has gone into making the buoy work well with alkaline
batteries rather than lithium-based batteries. Lithium batteries can deliver
much greater power and capacity, but mixing lithium batteries with water is
not safe (Hasvold et al., 2007). Since this buoy is designed to work close to
the coast and to be drifting freely there will always be an appreciable chance
that the buoy is found and picked up by the public. This puts stricter
requirements on power-use and maximum current draw.

We found the buoys lasted about $12$ days with continuous logging and
transmission during field experiments with a logging frequency of $52$ Hz,
storage to an SD card and frequent GPS fixes. This autonomy was obtained with
6 Alkaline C-cells with individual capacity of 8 Ah @ 1.5V (Duracell, 2019),
or 16 Ah @ 4.5 V with $2\times 3$ batteries. That amounts to an average
current use of about $\mathrm{16Ah/(12\times 24)h=0.055A=55mA}$. However, when
the buoy is configured to take positions every hour, without an SD-card, and
with a lower output frequency, the current drain is reduced down to about
$7.8$ mA (or $32.8$ mW).

Figure 17: Current measured using a _CurrentRanger_. The periods with the
lowest current happen when the buoy is logging the IMU continuously, while not
doing anything else. Higher current is drawn when the GPS or cellular
communication is active. Spikes seen at the start of new communication or GPS
periods are measurement errors due to input-range change from $\mu$A to mA in
the _CurrentRanger_.

A moored version of the buoy (as described in Section 4) with $3\times 21$
Alkaline D-cells (21 parallel $4.5$ V) shows an initial voltage drop of $2.47$
mV on average each day for the first $60$ days. The voltage drop of Alkaline
cells is steeper in the beginning and the end the battery capacity. A
conservative estimate of the battery capacity with a working voltage between
$4.621$ and $3.650$ V, yields a battery time of about $390$ days.

##  Appendix B Cellular network range

Figure 18 shows a histogram of the distance to cellular tower used by the
moored version of the buoy. Most of the time it uses a tower about $80$ km
away.

Figure 18: The distance to the cellular tower from the buoy.

[B] Data transmission

##  Appendix C IMU sampling and data collection

Due to chip-to-chip differences every IMU samples at a slightly different
frequency (typically within $\pm 10\%$). The actual sample rate can be
retrieved from the IMU registers (STMicroelectronics, 2021) (section 6.4).
However, since the data is accurately timestamped using GPS, we estimate the
true frequency accurately by the median time difference between every
1024-sample batch. This is done routinely to ensure correct sampling
frequency.

The MCU configures the IMU to record acceleration and rotation into its first-
in, first-out (FIFO) queue continuously, and then drains and processes the
samples at sufficiently frequent intervals for the FIFO to not fill up. The
MCU sleeps in between, saving power.

The data are collected in batches of 1024 samples and transmitted in messages
tagged with position and timestamps. These data are relayed to a server that
stores the messages in a database. A set of processing scripts read all
messages in a given range and concatenate the data and construct a continuous
time coordinate. The data are stored in CF-compliant NetCDF or zarr files with
all metadata, and should be fully self contained and ready for further
analysis.

[C] Schematic of mooring

##  Appendix D Fedjeosen mooring

Figure 19: Schematic of mooring configuration in Fedjeosen.

## References

  * Adafruit (2023) Adafruit, 2023: Adafruit AHRS library. Adafruit Industries.
  * Alari et al. (2022) Alari, V., and Coauthors, 2022: LainePoiss®—A Lightweight and Ice-Resistant Wave Buoy. J. Atmos. Oceanic Technol., 39 (5), 573–594, 10.1175/JTECH-D-21-0091.1.
  * Arraigada and Partl (2006) Arraigada, M., and M. Partl, 2006: Calculation of displacements of measured accelerations, analysis of two accelerometers and application in road engineering. 6th Swiss Transport Research Conference.
  * Babanin (2009) Babanin, A., 2009: Breaking of ocean surface waves. Acta Physica Slovaca. Reviews and Tutorials, 59 (4), 10.2478/v10155-010-0097-5.
  * Benetazzo (2006) Benetazzo, A., 2006: Measurements of short water waves using stereo matched image sequences. Coastal Engineering, 53 (12), 1013–1032, 10.1016/j.coastaleng.2006.06.012.
  * Boehm et al. (2017) Boehm, A. B., N. S. Ismail, L. M. Sassoubre, and E. A. Andruszkiewicz, 2017: Oceans in Peril: Grand Challenges in Applied Water Quality Research for the 21st Century. Environmental Engineering Science, 34 (1), 3–15, 10.1089/ees.2015.0252.
  * Bohlinger et al. (2019) Bohlinger, P., Ø. Breivik, T. Economou, and M. Müller, 2019: A novel approach to computing super observations for probabilistic wave model validation. Ocean Modelling, 139, 101 404, 10.1016/j.ocemod.2019.101404.
  * Breivik and Allen (2008) Breivik, Ø., and A. A. Allen, 2008: An Operational Search and Rescue Model for the Norwegian Sea and the North Sea. Journal of Marine Systems, 69 (1-2), 99–113, 10.1016/j.jmarsys.2007.02.010, 1111.1102.
  * Brown et al. (2018) Brown, A., J. Thomson, A. Ellenson, F. T. Rollano, H. T. Ozkan-Haller, and M. C. Haller, 2018: Kinematics and Statistics of Breaking Waves Observed Using SWIFT Buoys. IEEE J. Oceanic Eng., 44 (4), 1011–1023, 10.1109/JOE.2018.2868335.
  * Brown and Paasch (2021) Brown, A. C., and R. K. Paasch, 2021: The Accelerations of a Wave Measurement Buoy Impacted by Breaking Waves in the Surf Zone. JMSE, 9 (2), 214, 10.3390/jmse9020214.
  * Chuang et al. (2013) Chuang, L. Z.-H., L.-C. Wu, and J.-H. Wang, 2013: Continuous Wavelet Transform Analysis of Acceleration Signals Measured from a Wave Buoy. Sensors, 13 (8), 10 908–10 930, 10.3390/s130810908.
  * Collins et al. (2014) Collins, C. O., B. Lund, T. Waseda, and H. C. Graber, 2014: On recording sea surface elevation with accelerometer buoys: Lessons from ITOP (2010). Ocean Dynamics, 64 (6), 895–904, 10.1007/s10236-014-0732-7.
  * Datawell BV (2023) Datawell BV, 2023: Datawell Waverider Manual - DWR4. Tech. rep.
  * Deike (2022) Deike, L., 2022: Mass Transfer at the Ocean–Atmosphere Interface: The Role of Wave Breaking, Droplets, and Bubbles. Annual Review of Fluid Mechanics, 54 (1), 191–224, 10.1146/annurev-fluid-030121-014132.
  * Donelan et al. (1996) Donelan, M. A., W. M. Drennan, and A. K. Magnusson, 1996: Nonstationary Analysis of the Directional Properties of Propagating Waves. Journal of Physical Oceanography, 26 (9), 1901–1914, 10.1175/1520-0485(1996)026¡1901:NAOTDP¿2.0.CO;2.
  * Donelan et al. (1985) Donelan, M. A., J. Hamilton, and W. H. Hui, 1985: Directional spectra of wind-generated ocean waves. Phil. Trans. R. Soc. Lond. A, 315 (1534), 509–562, 10.1098/rsta.1985.0054.
  * Dorgeville et al. (2018) Dorgeville, E., H. Tholo, and D. A. Tengberg, 2018: Data validation for the MOTUS Directional Wave Buoy. Tech. rep.
  * Duracell (2019) Duracell, 2019: Duracell ProCell Intense (Datasheet). Tech. rep.
  * Edinburgh Designs (2023) Edinburgh Designs, 2023: Piston Coastal Wave Generators — Edinburgh Designs. http://www4.edesign.co.uk/product/piston-wave-generators/.
  * E.U. Copernicus Marine Service Information . Marine Data Store (MDS)(2023) (CMEMS) E.U. Copernicus Marine Service Information (CMEMS). Marine Data Store (MDS), 2023: Global Ocean L 3 Significant Wave Height From Nrt Satellite Measurements. 10.48670/moi-00179.
  * Feddersen et al. (2023a) Feddersen, F., A. Amador, K. Pick, A. Vizuet, K. Quinn, E. Wolfinger, J. H. MacMahan, and A. Fincham, 2023a: The wavedrifter: A low-cost IMU-based Lagrangian drifter to observe steepening and overturning of surface gravity waves and the transition to turbulence. Coastal Engineering Journal, 1–14, 10.1080/21664250.2023.2238949.
  * Feddersen et al. (2023b) Feddersen, F., A. M. Fincham, K. L. Brodie, A. P. Young, M. S. Spydell, D. J. Grimes, M. Pieszka, and K. Hanson, 2023b: Cross-shore wind-induced changes to field-scale overturning wave shape. Journal of Fluid Mechanics, 958, A4, 10.1017/jfm.2023.40.
  * Fisher et al. (2021) Fisher, A., J. Thomson, and M. Schwendeman, 2021: Rapid deterministic wave prediction using a sparse array of buoys. Ocean Engineering, 228, 108 871, 10.1016/j.oceaneng.2021.108871.
  * Forristall (2000) Forristall, G. Z., 2000: Wave Crest Distributions: Observations and Second-Order Theory. J. Phys. Oceanogr., 30 (8), 1931–1943, 10.1175/1520-0485(2000)030¡1931:WCDOAS¿2.0.CO;2.
  * Halsne et al. (2023) Halsne, T., A. Benetazzo, F. Barbariol, K. H. Christensen, A. Carrasco, and Ø. Breivik, 2023: Wave modulation in a strong tidal current and its impact on extreme waves. Journal of Physical Oceanography, 10.1175/JPO-D-23-0051.1.
  * Halsne et al. (2022) Halsne, T., P. Bohlinger, K. H. Christensen, A. Carrasco, and Ø. Breivik, 2022: Resolving regions known for intense wave–current interaction using spectral wave models: A case study in the energetic flow fields of Northern Norway. Ocean Modelling, 176, 102 071, 10.1016/j.ocemod.2022.102071.
  * Hasvold et al. (2007) Hasvold, Ø., S. Forseth, T. C. Johannessen, and T. Lian, 2007: Safety aspects of large lithium batteries. Tech. rep., FFI.
  * Holthuijsen (2010) Holthuijsen, L. H., 2010: Waves in Oceanic and Coastal Waters. Cambridge University Press.
  * Magnusson et al. (1999) Magnusson, A., M. Donelan, and W. Drennan, 1999: On estimating extremes in an evolving wave field. Coastal Engineering, 36 (2), 147–163, 10.1016/S0378-3839(99)00004-6.
  * Malila (2022) Malila, M., 2022: An investigation into the dynamical and statistical properties of dominant ocean surface waves using close-range remote sensing. Ph.D. thesis, University of Bergen, Bergen.
  * Malila et al. (2022) Malila, M. P., J. Thomson, Ø. Breivik, A. Benetazzo, B. Scanlon, and B. Ward, 2022: On the Groupiness and Intermittency of Oceanic Whitecaps. JGR Oceans, 127 (1), e2021JC017 938, 10.1029/2021JC017938.
  * Martini et al. (2021) Martini, A., R. Hoeke, R. Crossing, B. D. Groot, P. Branson, and T. Pitman, 2021: Mooring Design and Operational Guidelines for Lightweight Wave Buoys. Tech. rep., CSIRO, Australia.
  * Melville (1996) Melville, W. K., 1996: The Role of Surface-Wave Breaking in Air-Sea Interaction. Annu. Rev. Fluid Mech.
  * Nose et al. (2018) Nose, T., A. Webb, T. Waseda, J. Inoue, and K. Sato, 2018: Predictability of storm wave heights in the ice-free Beaufort Sea. Ocean Dynamics, 68 (10), 1383–1402, 10.1007/s10236-018-1194-0.
  * Nose et al. (2023) Nose, T., and Coauthors, 2023: arXiv:2302.02820. A comparison of an operational wave-ice model product and drifting wave buoy observation in the central Arctic Ocean: Investigating the effect of sea ice forcing in thin ice cover. arXiv, 2302.02820.
  * NXP Semiconductors (2016) NXP Semiconductors, 2016: NXP Sensor Fusion - NXP Sensor Fusion for Kinetis MCUs.
  * Rabault et al. (2020) Rabault, J., G. Sutherland, O. Gundersen, A. Jensen, A. Marchenko, and Ø. Breivik, 2020: An open source, versatile, affordable waves in ice instrument for scientific measurements in the Polar Regions. Cold Regions Science and Technology, 170, 102 955, 10.1016/j.coldregions.2019.102955.
  * Rabault et al. (2022) Rabault, J., and Coauthors, 2022: OpenMetBuoy-v2021: An Easy-to-Build, Affordable, Customizable, Open-Source Instrument for Oceanographic Measurements of Drift and Waves in Sea Ice and the Open Ocean. Geosciences, 12 (3), 110, 10.3390/geosciences12030110.
  * Rabault et al. (2023) Rabault, J., and Coauthors, 2023: A dataset of direct observations of sea ice drift and waves in ice. Sci. Data.
  * Raghukumar et al. (2019) Raghukumar, K., G. Chang, F. Spada, C. Jones, T. Janssen, and A. Gans, 2019: Performance Characteristics of “Spotter,” a Newly Developed Real-Time Wave Measurement Buoy. Journal of Atmospheric and Oceanic Technology, 36 (6), 1127–1141, 10.1175/JTECH-D-18-0151.1.
  * Rainville et al. (2023) Rainville, E., J. Thomson, M. Moulton, and M. Derakhti, 2023: Measurements of nearshore ocean-surface kinematics through coherent arrays of free-drifting buoys. Earth System Science Data, 15 (11), 5135–5151, 10.5194/essd-15-5135-2023.
  * Rainville (2022) Rainville, E. J., 2022: Measurements of Nearshore Waves through Coherent Arrays of Small-Scale, Free-Drifting Wave Buoys. MSc, University of Washington, Seattle.
  * Saetra et al. (2021) Saetra, Ø., T. Halsne, A. Carrasco, Ø. Breivik, T. Pedersen, and K. H. Christensen, 2021: Intense interactions between ocean waves and currents observed in the Lofoten Maelstrom. Journal of Physical Oceanography, 10.1175/JPO-D-20-0290.1.
  * Schwendeman and Thomson (2014) Schwendeman, M., and J. Thomson, 2014: Wave Breaking Dissipation in a Young Wind Sea in: Journal of Physical Oceanography. Journal of Physical Oceanography, 44 (1).
  * Seldal (2023) Seldal, T. I., 2023: SFY: A free-drifting wave buoy for measuring surf waves. M.S. thesis, The University of Bergen.
  * Sinclair (2014) Sinclair, A., 2014: FlowRider: A Lagrangian Float to Measure 3-D Dynamics of Plunging Breakers in the Surf Zone. Journal of Coastal Research, 293, 205–209, 10.2112/JCOASTRES-D-13-00014.1.
  * Singsaas et al. (2020) Singsaas, I., F. Leirvik, P. S. Daling, C. Guénette, and K. R. Sørheim, 2020: Fate and behaviour of weathered oil drifting into sea ice, using a novel wave and current flume. Marine Pollution Bulletin, 159, 111 485, 10.1016/j.marpolbul.2020.111485.
  * STMicroelectronics (2019) STMicroelectronics, 2019: LSM6DSOX (Datasheet). Tech. rep.
  * STMicroelectronics (2020) STMicroelectronics, 2020: ISM330DHCX (Datasheet). Tech. rep.
  * STMicroelectronics (2021) STMicroelectronics, 2021: AN5398: ISM330DHCX: Always-on 3D accelerometer and 3D gyroscope with digital output for industrial applications.
  * The Wamdi Group (1988) The Wamdi Group, 1988: The WAM Model—A Third Generation Ocean Wave Prediction Model. Journal of Physical Oceanography, 18 (12), 1775–1810, 10.1175/1520-0485(1988)018¡1775:TWMTGO¿2.0.CO;2.
  * Thomson (2012) Thomson, J., 2012: Wave Breaking Dissipation Observed with “SWIFT” Drifters. Journal of Atmospheric and Oceanic Technology, 29 (12), 1866–1882, 10.1175/JTECH-D-12-00018.1.
  * Thomson et al. (2018) Thomson, J., J. B. Girton, R. Jha, and A. Trapani, 2018: Measurements of Directional Wave Spectra and Wind Stress from a Wave Glider Autonomous Surface Vehicle. Journal of Atmospheric and Oceanic Technology, 35 (2), 347–363, 10.1175/JTECH-D-17-0091.1.
  * Thomson et al. (2015) Thomson, J., and Coauthors, 2015: Biofouling Effects on the Response of a Wave Measurement Buoy in Deep Water. Journal of Atmospheric and Oceanic Technology, 32 (6), 1281–1286, 10.1175/JTECH-D-15-0029.1.
  * Thomson et al. (2019) Thomson, J., and Coauthors, 2019: A new version of the SWIFT platform for waves, currents, and turbulence in the ocean surface layer. 2019 IEEE/OES Twelfth Current, Waves and Turbulence Measurement (CWTM), IEEE, San Diego, CA, USA, 1–7, 10.1109/CWTM43797.2019.8955299.
  * Thomson et al. (2023) Thomson, J., and Coauthors, 2023: Development and testing of microSWIFT expendable wave buoys. Coastal Engineering Journal, 1–13, 10.1080/21664250.2023.2283325.
  * Tucker (1958) Tucker, M., 1958: The accuracy of wave measurements made with vertical accelerometers. Deep Sea Research (1953), 5 (2-4), 185–192, 10.1016/0146-6313(58)90009-1.
  * Van Essen et al. (2018) Van Essen, S., K. Ewans, and J. McConochie, 2018: Wave Buoy Performance in Short and Long Waves, Evaluated Using Tests on a Hexapod. Volume 7B: Ocean Engineering, American Society of Mechanical Engineers, Madrid, Spain, V07BT06A001, 10.1115/OMAE2018-77092.
  * Veras Guimarães et al. (2018) Veras Guimarães, P., and Coauthors, 2018: A surface kinematics buoy (SKIB) for wave–current interaction studies. Ocean Sci., 14 (6), 1449–1460, 10.5194/os-14-1449-2018.
  * Waseda et al. (2018) Waseda, T., A. Webb, K. Sato, J. Inoue, A. Kohout, B. Penrose, and S. Penrose, 2018: Correlated Increase of High Ocean Waves and Winds in the Ice-Free Waters of the Arctic Ocean. Sci Rep, 8 (1), 4489, 10.1038/s41598-018-22500-9.
  * Yurovsky and Dulov (2020) Yurovsky, Y. Y., and V. A. Dulov, 2020: MEMS-based wave buoy: Towards short wind-wave sensing. Ocean Engineering, 217, 108 043, 10.1016/j.oceaneng.2020.108043.