# Ocean wave measurements for marine renewable energy applications

Daniel Peláez-Zapata, Vikram Pakrashi, Frédéric Dias

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Daniel Peláez-Zapata, Vikram Pakrashi, Frédéric Dias. Ocean wave measurements for marine renewable energy applications. Renewable and Sustainable Energy Reviews, 2025, 219, pp.115828. 10.1016/j.rser.2025.115828. hal-05468218

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Review article

Ocean wave measurements for marine renewable energy applications

Daniel Peláez-Zapata a,b ID,*, Vikram Pakrashi c ID, Frédéric Dias a,b ID

<figure><img src="image_3.png" /></figure>

ARTICLE INFO

Keywords:

MRE

Wave buoys

ADCPs

Hydrophones

Directional wave spectrum

WECs

Offshore wind farms

ABSTRACT

Accurate wave measurements are crucial for multiple aspects of marine renewable energy projects, including assessing environmental conditions, estimating wave-induced loads on fixed and floating platforms and validating numerical models. This review explores the current state of wave measuring techniques and sensors, their advantages, limitations, and recent technological advancements. Various approaches including *in-situ* measurements using wave buoys, acoustic Doppler current profilers (ADCPs) and hydrophones are discussed. Challenges in acquiring accurate and reliable measurements, effects of the marine environment on devices, and limitations in measuring technologies are highlighted. This review identifies future research areas including power-efficient long-term monitoring, directional wave characteristics, and better sensor integration. Addressing these challenges can lead to enhanced understanding of ocean waves, contributing to improved wave characterisation in the marine renewable energy sector. Additionally, this study aims to contribute towards sustainable development by advancing affordable and clean energy technologies and addressing the challenges of climate change.

Contents

<table><tr><td>1.</td><td>Introduction</td><td>3</td></tr><tr><td></td><td>1.1. Importance of wave measurements in the MRE industry</td><td>3</td></tr><tr><td></td><td>1.2. Wave data requirements for MRE projects</td><td>3</td></tr><tr><td></td><td>1.3. Challenges and opportunities in measuring ocean waves</td><td>3</td></tr><tr><td></td><td>1.4. Background and related works</td><td>3</td></tr><tr><td></td><td>1.5. Objectives, novelty and contribution</td><td>3</td></tr><tr><td></td><td>1.6. Scope and limitations</td><td>4</td></tr><tr><td></td><td>1.7. Structure of this review</td><td>5</td></tr><tr><td>2.</td><td>Standard mathematical framework</td><td>5</td></tr><tr><td>3.</td><td>Wave buoys</td><td>6</td></tr><tr><td></td><td>3.1. Principles of operation</td><td>7</td></tr><tr><td></td><td>3.2. Capabilities and limitations</td><td>8</td></tr><tr><td></td><td>3.3. Shape, size and payload</td><td>9</td></tr><tr><td></td><td>3.4. Comparison of wave measurements</td><td>9</td></tr><tr><td></td><td>3.5. Other applications</td><td>9</td></tr><tr><td></td><td>3.6. Recent developments in wave buoy technology</td><td>10</td></tr><tr><td>4.</td><td>Acoustic Doppler current profilers</td><td>10</td></tr><tr><td></td><td>4.1. Principles of operation</td><td>10</td></tr><tr><td></td><td>4.2. Processing techniques</td><td>10</td></tr><tr><td></td><td>4.3. Range, resolution and random noise trade-off</td><td>11</td></tr><tr><td></td><td>4.4. ADCP geometry and velocity estimation</td><td>11</td></tr><tr><td></td><td>4.5. Wave measurements</td><td>12</td></tr></table>

<footnote>ª Centre Borelli, École normale supérieure Paris-Saclay, 4 Av. des Sciences, Gif-sur-Yvette, 91190, France</footnote>

<footnote>ᵇ School of Mathematics and Statistics, University College Dublin, Dublin, D04V1W8, Ireland</footnote>

<footnote>ᶜ UCD Centre for Mechanics, Dynamical Systems and Risk Laboratory, School of Mechanical and Materials Engineering, University College Dublin, Dublin, D04V1W8, Ireland</footnote>

<footnote>* Corresponding author at: Centre Borelli, École normale supérieure Paris-Saclay, 4 Av. des Sciences, Gif-sur-Yvette, 91190, France.</footnote>

<footnote>E-mail address: daniel.pelaez_zapata@ens-paris-saclay.fr (D. Peláez-Zapata).</footnote>

<footnote>URL: https://dspelaez.github.io (D. Peláez-Zapata).</footnote>
# 1. Introduction

## 1.1. Importance of wave measurements in the MRE industry

Ocean wave measurements have been crucial for industry applications for several decades. One of the best known examples is the development of offshore platforms in the North Sea. Moreover, the scientific community has learnt a lot about waves thanks to the various sensors available on the platforms. As pointed out in Dudley et al. [1], it is now accepted that the study of rogue waves began with the measurement, on 1 January 1995, of a wave with trough-to-crest height of 25.6 m on the Draupner oil platform in the North Sea [2]. The fact that this wave was observed within an extended time series at high-sampling frequency yielded great confidence in the measurement fidelity and stimulated the oceanographic community to investigate the physics and statistics of ocean rogue waves on a quantitative level. With the rapid advancement of marine renewable energy (MRE) technologies, accurate and continuous in-situ wave measurements are vital in harnessing the immense power of the sea, contributing towards a cleaner and greener future.

## 1.2. Wave data requirements for MRE projects

The development of MRE projects relies on diverse data sources including numerical models, remote sensing, and in-situ measurements. More precisely, wave data requirements may vary according to the particular MRE technology and serve specific purposes at different project stages (Fig. 1). Probably the most illustrative example of wave data requirements during the planning phase of wave energy converters (WECs) is the resource assessment [3–6]. Long-term wave statistics and spatial information are important for selecting the optimal installation site, estimating the power production and evaluating the device performance [7,8]. This step is typically supported by spectral wave models [9,10], computational fluid dynamics (CFD) tools [11,12], and hindcast databases [13–15]. In-situ wave measurements are therefore crucial in calibrating, validating and enhancing these models for site-specific conditions.

Analysis of extreme wave data is required at the design stage for ensuring the operability and survivability of not only WECs [16] but also offshore wind farms [17–19], as well as any other type of fixed or floating structure in the marine environment. In general, continuous wave monitoring ensures accurate characterisation of wave conditions, optimising device design, assessing environmental impacts, ensuring safe operations, maximising energy production, and facilitating accurate resource assessment and decision-making processes in the MRE industry.

Each MRE technology has unique interplays with wave conditions, requiring specific data to ensure optimal performance (Table 1). For example, studies have shown that swell modifies the vertical wind shear in the boundary layer [20]. This can have implications for turbulence intensity, an important design parameter for offshore wind turbines [21,22]. Additionally, the wind-wave misalignment also plays a significant role in the turbine design and operation [23]. In both cases, the directional wave spectrum is necessary to separate the contribution of swells from the local wind-waves.

Another example is the impact of wave-induced motion on the amount of sunlight, and consequently, on the energy production of offshore floating photovoltaic farms [24,25]. Additionally, waves can enhance the mixing of the upper ocean either by direct injection of turbulence [26] or by Langmuir circulation [27–29]. This may impact the design of ocean thermal energy conversion (OTEC) projects [30]. While certain coastal MRE technologies, such as salinity gradient energy (SGE) at river mouths [31,32], might not be directly influenced by waves, the effects of wave action in the surf-zone should generally be taken into consideration.

## 1.3. Challenges and opportunities in measuring ocean waves

Understanding working principles, capabilities, and limitations of wave sensors is essential for accurately describing wave properties. Nevertheless, measuring ocean waves represents a significant challenge due to several factors. First, the size and scale of the ocean make it difficult to acquire measurements across large spatial scales and for extended periods of time. Second, the harsh marine environment can be damaging to measurement devices, which must be designed to withstand the corrosive effects of saltwater and strong currents. Third, there are limitations in measuring technologies, including the trade-off between measurement accuracy and cost, and the difficulty of measuring waves in open ocean. Finally, the inherent complexity and variability of wave dynamics make it challenging to obtain reliable measurements.

Despite these challenges, various approaches exist for measuring ocean waves. While the most extended methods rely on analysing the kinematics of wave motion, there are also other interesting aspects that can be exploited, including the interaction between microwaves and the sea surface roughness, as well as the acoustic properties of the ocean upper layer in the presence of waves (Fig. 2).

The most common techniques include in-situ point measurements, such as wave buoys and wave gauges [33–36]; on-shore observation stations such as stereo video-cameras [37–39] or marine radars [40–42]; remote sensing techniques such as satellite-mounted radars [43, 44] and airborne-mounted lidar systems [45–47]; and acoustic methods such as ADCPs [48,49] and hydrophones [50–52].

Each of these sensors presents distinct advantages and limitations in terms of accuracy, precision, reliability, spatial coverage, cost, and overall performance. Consequently, an in-depth literature review on wave measurement in the context of MRE is essential to consolidate the current state of knowledge and identify the most effective measurement techniques and technologies.

## 1.4. Background and related works

Recent literature reviews have investigated different methodologies for in-situ wave measurements with broad applicability. For instance, Rossi et al. [53] examined the current state of development of various techniques for measuring ocean waves, discussing the need for observation networks that work together and integrate different measurement techniques to effectively describe the sea state. Rao et al. [54] collected and analysed publications from 1970 to 2017 regarding wave measurements using ADCPs, providing an overview of the evolution of these sensors. Ardhuin et al. [55] discussed the importance of sea state information for various applications and highlighted the limitations of current observation systems, concluding that remote sensing and in-situ measurements can be combined with numerical wave models to achieve the resolution at kilometre scales. Furthermore, ongoing developments in satellite observing systems have extended the capabilities of remote sensing and in-situ measurements, with an emphasis on providing forcing and validation data for numerical wave models, particularly directional wave spectra, rather than solely relying on bulk wave parameters [44].

## 1.5. Objectives, novelty and contribution

This review seeks to explore the current state of wave sensors and measuring techniques, including their underlying working principles, advantages, limitations, recent technological advancements and overall performance. Additionally, the paper aims to highlight the challenges in acquiring accurate and reliable wave measurements for MRE projects, while identifying future research areas for improved understanding and wave characterisation. While this review does not adhere to a systematic protocol, it aims to synthesise and critically analyse the most relevant body of knowledge on ocean wave measurements and
<figure><img src="image_4.png" /><figcaption>Fig. 1. Broad examples and applications of wave data requirements during the lifetime of a typical MRE project.</figcaption></figure>

<caption>Table 1</caption>

Summary of wave data requirements for different MRE technologies. The table highlights the purpose of these data requirements and a representative instrumentation, which is not exclusive and is offered merely as an illustrative example.

<table><thead><tr><th>MRE technology</th><th>Wave data requirement</th><th>Purpose/Use</th><th>Representative instrumentation</th></tr></thead><tbody><tr><td rowspan="2">WECs</td><td>Average and extreme wave parameters</td><td>Optimise energy capture and device design</td><td>Long-term monitoring using large buoys or bottom-mounted ADCP</td></tr><tr><td>Directional wave spectrum</td><td>Align devices optimally</td><td>Wave staff array complemented with high-resolution remote sensing such as cameras or radars</td></tr><tr><td rowspan="2">Offshore Wind</td><td>Wave-wind interaction and spectral partitions</td><td>Turbine performance and downtime reduction</td><td>MetBuoy (with anemometer) in combination with bottom-mounted ADCP</td></tr><tr><td>Wind-wave misalignment</td><td>Ensure turbine stability and optimal operation</td><td>Spar buoy equipped with sonic anemometers</td></tr><tr><td rowspan="2">Offshore Solar</td><td>Wave-induced tilt and drift</td><td>Optimise sunlight exposure and energy production</td><td>Wave buoy or bottom-mounted ADCP with real-time data transmission</td></tr><tr><td>Platform survivability</td><td>Ensure durability in offshore conditions</td><td>Robust small buoys with integrated GPS/IMU</td></tr><tr><td rowspan="2">Tidal/Currents</td><td>Wave-current interaction</td><td>Understand wave influences on turbine performance</td><td>Bottom-mounted ADCP to obtain currents profile</td></tr><tr><td>Wave-induced turbulence</td><td>Protect turbine blades and moorings</td><td>High-resolution ADCP to obtain TKE estimations</td></tr><tr><td>OTEC</td><td>Wave-induced mooring loads</td><td>Maintain stability of floating platforms</td><td>Collected wave buoy data can be used to feed CFD models and tank experiments</td></tr></tbody></table>

their implications for the MRE industry, providing an interdisciplinary perspective on this subject and to highlighting emerging topics.

The novelty of the present review lies in its specific focus on wave sensors for MRE applications, unlike previous studies that discussed wave measurements in a broad way. This paper fills a gap in the existing literature by unifying scattered information on wave measurements, providing a valuable resource for the MRE scientific and industry community. The content and findings presented in this paper are expected to contribute towards sustainable development in alignment with the United Nations Sustainable Development Goals (SDGs), specifically SDG 7 (Affordable and Clean Energy) and SDG 13 (Climate Action).

The research relevant questions addressed here are: What are the underlying principles of each sensor and how do they work?

What information can be extracted from each sensor and what analysis techniques are commonly used to obtain meaningful information? What are the recent technological developments and advancements in each sensor type and their implications for improving wave monitoring in MRE projects, what is the particular role of wave breaking and wave directionality in the context of MRE?

## 1.6. Scope and limitations

This study focuses particularly on wave buoys, ADCPs and hydrophones. The emphasis on ADCPs and wave buoys is justified as both instruments use a common mathematical framework, allowing for consistent data analysis and interpretation. Additionally, these are commercially available devices that can be easily acquired by researchers
<figure><img src="image_5.png" /><figcaption>Fig. 2. Illustration of typical sensors and instruments to obtain ocean waves information, highlighting the use of various aspects of ocean waves, such as kinematic, acoustics and microwave interactions. The diagram includes land-based sensors such as video-cameras and radars; bottom-mounted upward-looking ADCPs capable of measuring waves, currents, and turbulence; a satellite employing microwave sensing for ocean wave data retrieval and data transmission; and a typical surface-following wave buoy, with a hydrophone attached to demonstrate buoy-mounted payload.</figcaption></figure>

and engineers. Hydrophones, on the other hand, are commonly used in the MRE industry for environmental noise control. However, their potential to measure wave characteristics makes them a valuable addition to this review. A detailed review on remote sensing techniques, such as stereo-imaging, radars systems and satellite-based measurements is beyond the scope of this paper. However, given their importance within wave observations techniques, the most relevant aspects are briefly discussed.

## 1.7. Structure of this review

This document is structured as follows: an overview of the importance of wave measurements in the context of MRE is presented in Section 1. The standard mathematical framework and definition of wave parameters is presented in Section 2. Sections 3 to 5 discuss the principle of operation, capabilities and limitations and technological advances of wave buoys, ADCPs and hydrophones, respectively. Section 6 briefly highlights the main aspects of remote sensing techniques, including cameras, radars and satellite-based wave measurements. A discussion on critical aspects of wave measurements, including challenges, economic considerations, regulations and future research directions is offered in Section 7. Finally, Section 8 offers a summary of the review, highlighting the more important findings.

## 2. Standard mathematical framework

This section presents the conventional mathematical framework for describing wave motion and quantifying wave parameters. Specifically, the focus is on parameters obtained from the directional wave spectrum that are of practical use such as significant wave height, peak period, wave direction and directional spreading. This framework is valid for measurements taken from typical in-situ sensors, including wave buoys, wave staffs arrays, ADCPs or pressure sensors.

Although ocean waves are inherently non-linear, a linear representation often provides a reliable approximation under typical conditions, particularly in deep waters and for moderate wave steepness. Wave measurements can always be interpreted in a non-linear context, such as infragravity waves directionality [56], time-frequency decomposition [57] and bi-spectral analysis [58]. However this should be done with some caution (see Section 7). For practical purposes, only the linear approximation is presented here. Under this assumption, the sea

surface elevation $η(x, t)$ can be described as a superposition of plane waves with different scales, directions and phases [59–63]. This allows for a convenient representation in the wavenumber-frequency domain using the Fourier transform:

$$η(x, t) = \int_{-\infty}^{\infty} Z(k, \omega)e^{i(k \cdot x - \omega t)} dk d\omega, \quad (1)$$

where $Z(k, \omega)$ are the complex Fourier coefficients that contains the amplitude and phase of wave components with wavenumber $k$ and angular frequency $\omega$. The result of multiplying the Fourier coefficients by its complex conjugate is the full three-dimensional wave spectral energy density:

$$E(k, \omega)dkd\omega = Z(k, \omega)Z^*(k, \omega), \quad (2)$$

where * denotes complex conjugate. This function provides a complete description of the wave energy density distribution across a range of frequencies, wavenumbers and directions. The frequencies and wavenumbers are related through the dispersion relation

$$\sigma^2 = gk \tanh kh, \quad (3)$$

where $g$ is the acceleration due to gravity, $k = |k|$ is the wavenumber magnitude, $h$ is the water depth and $\sigma$ represents the relative angular frequency, measured in a frame of reference moving with the ocean currents [55]. On the other hand, the absolute frequency $\omega$, measured in a frame of reference fixed to the Earth is given by

$$\omega = \sigma + k \cdot U, \quad (4)$$

where $U$ represents the advection caused by ocean currents, which is a function of the wavenumber.

Drifting wave buoys measure the relative frequency as they travel with ocean currents, whereas fixed-point sensors such as moored wave buoys and bottom-mounted ADCPs, observe the Doppler-shifted absolute wave frequency. This may seriously affect the interpretation of the wavelength scales, particularly for shorter waves [64,65]. However, this effect becomes more important for currents speeds greater than 0.2 m s⁻¹ [55].

The dispersion relation is generally used to write the full wave spectrum in terms of the more commonly used frequency-direction spectrum $E(\omega, \theta)$. The most conventional wave quantities used in physical oceanography to describe the wave field are derived from the
directional spectrum. The significant wave height is calculated as [66]:

$$H_s = 4 \left[ \iint E(\omega, \theta) d\omega d\theta \right]^{1/2} . \qquad (5)$$

There are several definitions of the peak wave period, however the most widely accepted was proposed by Young [67]:

$$T_p = 2\pi \frac{\iint E^4(\omega, \theta) d\omega d\theta}{\iint \omega E^4(\omega, \theta) d\omega d\theta} . \qquad (6)$$

In a general way, a combination of three orthogonal wave-related quantities is sufficient to yield an acceptable description of the wave direction [62]. For instance, heave-pitch-roll buoys are able to measure vertical acceleration and orthogonal sea surface slopes, while GPS buoys directly deliver the sea surface elevation and the two perpendicular horizontal wave displacements or velocities. Other wave measurement systems, such as ADCPs, measure the three orthogonal components of the orbital motions. Regardless the measured quantity, the cross-spectrum links the instrument measurements and the directional wave spectrum. The definition of the cross-spectrum between any pair of wave-related quantities (slope, velocities, acceleration, pressure, etc.), located at the points $x_m$ and $x_n$, respectively, can be written as:

$$\Phi_{mn}(\omega) = \int_{-\pi}^{\pi} H_m(\omega, \theta) H_n^*(\omega, \theta) e^{ik \cdot (x_m - x_n)} E(\omega, \theta) d\theta, \qquad (7)$$

where $H(\omega, \theta)$ is a transfer function that relates the measured wave quantity and the wave surface elevation. The directional wave spectrum is usually written as $E(\omega, \theta) = S(\omega)D(\omega, \theta)$, where $S(\omega)$ is the azimuthally-integrated frequency spectrum and $D(\omega, \theta)$ is the directional distribution function, which is subject to the following constraint:

$$\int_{-\pi}^{\pi} D(\omega, \theta) d\theta = 1. \qquad (8)$$

The exponential term in Eq. (7) is specifically relevant for analysing spatially distributed measurements, such as arrays of wave staffs [68] or ADCP along-beam velocities [49,69,70]. This term considers the geometric separation between individual sensors and the characteristics of the incident waves being measured. The performance of a spatial array is influenced not only by the number of sensors but also by their geometrical separation and the wavelength and direction of the incident waves [61]. On the other hand, surface-following buoys may be considered as a single-point system, i.e., all the quantities are measured approximately at the same spatial location ($x_m \approx x_n$), meaning that the exponential term in Eq. (7) is not relevant.

The form of the transfer function $H(\omega, \theta)$ depends on the type of measuring instrument being used. It is generally derived from the wave geometry or the wave kinematics using the linear wave theory. For example, buoy tilting, known as pitch and roll, can be essentially understood as the sea surface slopes, or in other words, as the spatial derivatives of the surface elevation in x and y directions, respectively. Table 2 summarises the most frequently used wave-related quantities and their corresponding transfer functions, outlining their definitions and mathematical expressions.

A combination of three time series produces a 3 × 3 cross-spectral matrix $\Phi_{mn}(\omega)$. Each element of the cross-spectral matrix is a complex quantity. For instance, in the particular case of the wave-induced buoy displacements, the cross-spectrum of the x and y variables is $S_{xy} = C_{xy} + iQ_{xy}$, where the real part is known as co-spectrum and the imaginary part as quad-spectrum. In addition, the auto-spectrum is a real quantity given by the correlation of the signal and its complex conjugate, and can be written as $E_{xx} = S_{xx}S_{xx}^*$. All these definitions are used to express the quantities that define the directional parameters of the wave spectrum. These are known as Fourier coefficients or circular moments of the directional distribution. Following the example of wave-induced displacements obtained by a GPS buoy, the first pairs of Fourier coefficients are obtained as:

$$a_1(\omega) = \int_{-\pi}^{\pi} D(\omega, \theta) \cos \theta d\theta = \frac{Q_{xz}}{\sqrt{E_{zz}(E_{xx} + E_{yy})}}, \qquad (9)$$

and

$$b_1(\omega) = \int_{-\pi}^{\pi} D(\omega, \theta) \sin \theta d\theta = \frac{Q_{yz}}{\sqrt{E_{zz}(E_{xx} + E_{yy})}}. \qquad (10)$$

Analogously, the second pair of Fourier coefficients are:

$$a_2(\omega) = \int_{-\pi}^{\pi} D(\omega, \theta) \cos 2\theta d\theta = \frac{E_{xx} - E_{yy}}{E_{xx} + E_{yy}}, \qquad (11)$$

and

$$b_2(\omega) = \int_{-\pi}^{\pi} D(\omega, \theta) \sin 2\theta d\theta = \frac{2Q_{xy}}{E_{xx} + E_{yy}}. \qquad (12)$$

The frequency-dependent wave direction can be estimated using the first and second pairs of Fourier coefficients as:

$$\theta_1(\omega) = \tan^{-1} \left( \frac{b_1}{a_1} \right) \qquad (13)$$

and

$$\theta_2(\omega) = \frac{1}{2} \tan^{-1} \left( \frac{b_2}{a_2} \right), \qquad (14)$$

respectively, and their corresponding directional spreading as:

$$\sigma_1(\omega) = \sqrt{2 \left( 1 - \sqrt{a_1^2 + b_1^2} \right)} \qquad (15)$$

and

$$\sigma_2(\omega) = \sqrt{\frac{1}{2} \left( 1 - \sqrt{a_2^2 + b_2^2} \right)}. \qquad (16)$$

In summary, the standard mathematical framework used for describing directional wave motion, based on in-situ measurements, has been presented. This framework is valid for either single-point measurements, such as wave buoys or ADCPs, or spatially distributed measurements, such as wave staff arrays or pressure sensors. Additionally, the most relevant wave parameters have been defined, namely significant wave height, peak wave period, wave direction and directional spreading, and their relationship with the instrument measurements has been described.

### 3. Wave buoys

Wave buoys have been one of the most widely used devices to perform wave measurements. They are characterised by their capability to withstand harsh ocean conditions and their ease of deployment. In addition, buoys can remain deployed for long periods, providing a continuous record of wave conditions over a specific region of the ocean. Furthermore, wave buoys can also be utilised as a versatile platform for deploying multiple sensors, enabling the acquisition of various environmental variables across the air-sea interface in addition to measuring wave characteristics. However, one of the main drawbacks is that buoys are limited to measuring only at single points, meaning that they cannot provide a complete picture of the wave field over a large area.

Wave buoys are typically anchored to the seafloor by one or several mooring lines, but they can also be freely drifting. Generally speaking, wave buoys can be classified into several types based on their measuring technology: IMU-based (acceleration and heave-pitch-roll); GPS buoys; and buoy-mounted wave staffs. Fig. 3 illustrates various types of buoys that have been traditionally used, not only to routinely measure wave parameters, but also as floating research platforms. This section introduces the theoretical basis of buoy-based wave measurements and presents an overview of the different types of wave buoys, as well as a comparison in terms of their ability to measure wave field properties.
<caption>**Table 2**</caption>

Summary of the most common wave-related quantities and their corresponding transfer functions. Here, z is the vertical coordinate measured from the mean sea level and positive upwards, $\eta$ denotes the surface elevation, $\phi$ is the velocity potential, $\alpha_n$ is the ADCP beam orientation and $\beta$ is vertical tilt of the ADCP. The information presented in this table is primarily based on the work by Young [61].

<table><thead><tr><th>Wave quantity</th><th>Symbol</th><th>Transfer function H(ω, θ)</th></tr></thead><tbody><tr><td>Surface elevation</td><td>η(x, t)</td><td>1</td></tr><tr><td>Surface slope in x (roll) and y (pitch)</td><td>∂η/∂x, ∂η/∂y</td><td>-ik cos θ, -ik sin θ</td></tr><tr><td>Surface curvatures in x, y and xy</td><td>∂²η/∂x², ∂²η/∂y², ∂²η/∂x∂y</td><td>-k² cos² θ, -k² sin² θ, -k² cos θ sin θ</td></tr><tr><td>Horizontal particle displacements</td><td>ξ<sub>x</sub>, ξ<sub>y</sub></td><td>-i (cosh k(z + h) / sinh kh) cos θ, -i (cosh k(z + h) / sinh kh) sin θ</td></tr><tr><td>Horizontal particle velocities</td><td>u = ∂φ/∂x, v = ∂φ/∂y</td><td>ω (cosh k(z + h) / sinh kh) cos θ, ω (cosh k(z + h) / sinh kh) sin θ</td></tr><tr><td>Horizontal particle accelerations</td><td>a<sub>x</sub> = ∂u/∂t, a<sub>y</sub> = ∂v/∂t</td><td>-iω² (cosh k(z + h) / sinh kh) cos θ, -iω² (cosh k(z + h) / sinh kh) sin θ</td></tr><tr><td>Vertical particle displacements</td><td>ξ<sub>z</sub></td><td>sinh k(z + h) / sinh kh</td></tr><tr><td>Vertical particle velocity</td><td>w = ∂φ/∂z</td><td>-iω (sinh k(z + h) / sinh kh)</td></tr><tr><td>Vertical particle acceleration</td><td>a<sub>z</sub> = ∂w/∂t</td><td>-ω² (sinh k(z + h) / sinh kh)</td></tr><tr><td>Pressure</td><td>p(x, z, t)</td><td>ρg (cosh k(z + h) / sinh kh)</td></tr><tr><td>ADCP along-beam velocities</td><td>b<sub>n</sub>(x, z, t)</td><td>ω / sinh kh [cosh kz cos(θ - α<sub>n</sub>) sin β - i sinh kz cos β]</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Examples of various buoy designs employed for wave measurements, illustrating the diversity of shapes, scales and functionalities tailored to specific data acquisition requirements.</figcaption></figure>

## 3.1. Principles of operation

### IMU-based buoys

For several decades, wave buoys have effectively used a combination of accelerometers, gyroscopes, and magnetometers as their primary sensing suite. In fact, the well-known Datawell Directional Waverider

(DWR) buoy, that incorporates these kind of sensors on-board, has become the industry standard for wave measurements [34,71–73].

The combination of these instruments, commonly referred to as Inertial Measurement Unit (IMU), provides a means of tracking the three-dimensional buoy position, velocity and orientation. Historically, a variety of technologies have been employed, including mechanical
sensors, quartz, and more recently, Micro-electromechanical Systems (MEMS) technology. As their name suggests, MEMS are composed by micro-components, such as beams or cantilevers, which generate an electrical signal when they move in response to acceleration or rotation. This technology has been widely adopted in the last decades due to its compactness, lightweight and cost-effectiveness, allowing for greater versatility and ease of integration into various measurement platforms [74–76].

The data acquired from IMUs typically includes linear acceleration along three orthogonal axes: surge (x-axis), sway (y-axis), and heave (z-axis), as well as the rate of change of buoy rotation along these axes: roll (x-axis), pitch (y-axis), and yaw (z-axis). Consequently, these systems are often referred to as heave-pitch-roll buoys. Additionally, IMUs use magnetometers to provide the orientation in relation to north, commonly referred to as heading. The wave-induced velocities and displacements can be approximated by integrating linear accelerations over time. Similarly, time-integration of the rotational changes provides a description of the buoy orientation. This integration is generally carried out in the frequency domain as a direct way to filter out undesired frequencies [77,78]. Directional wave spectrum is therefore obtained using conventional cross-spectral analysis, considering the transfer functions presented in Table 2.

## GPS-based buoys

Wave buoys incorporating GPS systems measure the Doppler-shifted radial velocity signal of the buoy as it responds to passing waves. The ephemerides provide the position and speed of the satellite orbit at all times. The buoy motion is then derived from the difference between the Doppler shift velocity and the satellite's known radial velocity. Given that these measurements merely produce radial velocities, a satellite in a near-nadir position can only approximate the buoy vertical velocity, whereas a satellite positioned closer to the horizon can provide information on the buoy horizontal motion. Consequently, a minimum of four satellites are required simultaneously to compute the buoy's velocity in the longitude, latitude and vertical coordinates [79].

Some systems, as the pioneer *Smart-800* proposed by Krogstad et al. [79], incorporated land-based stations in order to perform atmospheric corrections and hence obtain more precise Doppler-shifts. This makes these systems suitable only for coastal applications (within 10 km from shore). To overcome these limitations, single-GPS receivers were further introduced in the Datawell GPS version, the so-called DWR-G [71]. Wave orbital velocities are further integrated over time to obtain the three-dimensional wave-induced displacements.

## Wave staffs

Wave staffs typically consist of a series of vertically aligned semi-submerged wires that detect sea surface level by measuring changes in electrical properties between their exposed and submerged portions. These electrical properties can be either resistance [80] or capacitance [81]. In particular, capacitance wires have been widely used and normally consist of a conductive cylindrical centre surrounded by a dielectric material, typically “Teflon”. As the sea water level rises or falls, it changes the dielectric constant, causing a corresponding change in capacitance. With a proper calibration, this chance in capacitance can be translated directly to changes in sea level, with an accuracy of the order of 1 mm [82]. A single wire can provide information of the frequency wave spectrum, however arrays of wave staffs provide an alternative approach to acquiring accurate directional wave spectra information [61].

## 3.2. Capabilities and limitations

Accurate estimation of buoy inclination is crucial for obtaining precise directional information [62]. Nevertheless, the estimation of buoy inclination presents a considerable challenge when using IMU

buoys. The limitations of gyroscopes and accelerometers are well-known in the fields of robotics and navigation [83]. Gyroscopes, which measure angular velocities, are more reliable at higher frequencies as they can effectively capture rapid changes in orientation. However, they present difficulties with lower frequencies due to issues such as numerical drift. On the other hand, accelerometers are effective at measuring buoy inclinations at lower frequencies, accurately capturing steady or slowly varying forces. Nonetheless, at higher frequencies, they can be affected by vibrations and other transient forces, which introduces signal noise. A common solution to this problem is using simple signal fusion techniques, such as complementary filters [84], or more complex algorithms such as Kalman and Extended Kalman filters [83,85].

According to Herbers et al. [86], one of the evident advantages of GPS buoys over the traditional IMU systems is that buoy velocities can be determined directly from external GPS signals, eliminating moving parts and thus the need for on-board calibration. In addition, GPS buoys can be more cost-effective compared to traditional wave measurement methods. They require fewer maintenance and operational costs as they do not need regular deployments and retrievals. Nevertheless, GPS buoys present a critical limitation when water washing-over of the GPS antenna, interrupts the GPS satellite signals and thus corrupts the wave measurement process. However, de Vries et al. [71] reports that those incidents occurred less than a few seconds per day, even during significant wave heights of up to 4 m. In addition, as these buoys rely on satellite signals, limited or obstructed satellite visibility, such as in coastal areas or during severe weather conditions, can affect the accuracy and availability of data [35,71,86,87].

Wave staff arrays have been extensively used in laboratory experiments [82], near-shore bottom-fixed platforms [80–82] and moving ships [78]. In addition, wave staffs mounted on floating platforms such as buoys, have demonstrated optimal performance in resolving wave directional characteristics, particularly at high frequencies [82,88]. In contrast, Young [61] concluded that wave buoys, whether IMU or GPS-based, have limited ability to resolve the wave direction. To enhance instrument performance, they recommend to increase the elements in the array and proposed an optimal layout.

The ASIS (Air-Sea Interaction Spar) buoy, developed by Drennan et al. [89] and Graber et al. [68], is a notable example of spar buoy structures incorporating wave staffs. These buoys have proven successful in assessments carried out in the Gulf of Mexico [68], the Gulf of Tehuantepec [90], and Baja California [77]. The BOMM (Boya Oceanográfica y de Meteorología Marina) mirrors the ASIS design. Their overall good performance can be attributed to their robust design, high stability, and capability for conducting various meteorological and underwater measurements. In addition, wave staffs arrays can be used to obtain the full three-dimensional wave spectrum in terms of both wavenumber and frequency [88], which is of particular importance for air-sea and wave-current interaction studies.

Another comparable yet more compact design is presented in Pascal et al. [91], which has found utility in detecting breaking waves, underwater bubble size distributions and near-surface currents [92,93]. It is important to note that buoy-mounted wave staffs data are measured within a frame of reference relative to the buoy motion, which means that they primarily capture the higher frequency components of the wave spectrum. To accurately capture the full wave spectrum, it is necessary to combine wave staffs data with IMU measurements to account for the buoy motion, position and inclination [89].

An inherent limitation of moored buoys is the potential influence of the mooring dynamics on the measurements, especially for large waves. The mooring line, which anchors the buoy to the seabed, can restrict the buoy's motion, affecting the estimation of wave direction [94,95]. Laboratory experiments found that while the mooring had no major impact on vertical displacement, it had a notable effect on lateral buoy displacements, which are crucial for directional estimation [96]. To mitigate these effects, some designs incorporate compliant elastic
elements in the mooring, allowing the buoy to move more freely with surface waves [97]. Moreover, configurations such as the tether system designed for ASIS [68,89] have been employed to reduce this effect. Despite these improvements, accurately capturing the characteristics of large waves remains challenging. Non-invasive systems such as ADCPs (see Section 4) have demonstrated good performance in measuring extreme waves [98].

### 3.3. Shape, size and payload

There is a considerable range of sizes and shapes among the buoys being examined here. Buoys shapes can include spherical, spar, discus, or boat-shaped hull [55]. These characteristics can impact their performance and capabilities. The hull shape is clearly important when resolving short wave components [99]. For instance, the NDBC 3 m discus buoy cannot resolve waves shorter than 3 s [89]. Various Datawell DWR buoys with spherical hulls in three different sizes (0.9, 0.7, and 0.4 m diameter) were evaluated by Herbers et al. [86] in drifting conditions. Their findings indicated that the smaller diameter buoys moved in close proximity, whereas the larger buoy exhibited separation, implying a potential wind drag effect attributed to its larger size. Further buoy comparisons were presented by Jensen et al. [73]. The size of the buoy is also important for the potential influence of biological growth, which can subsequently affect hydrodynamic forces acting on the buoy [100]. How this affects the quality of wave measurements and the longevity of buoy deployment remains unclear.

Larger buoys generally offer advantages in terms of stability and buoyancy, making them more suitable for harsh weather conditions and rough seas [34,68,89]. Larger hulls also allow for increased storage capacity, enabling longer deployment and reducing the need for frequent maintenance visits. Large buoys typically have more internal space, providing opportunities for integrating additional instruments and equipment such as data loggers, power systems, and communication devices. These are indispensable for real-time transmission of large volumes of data, allowing for long-term monitoring.

For instance, the UK National Oceanography Centre (NOC) spar buoy includes instruments such as fibre-optic sensors, hydrophones, and acoustic velocimeters [91]. Additional sensors dedicated to measuring bubble characteristics were added in recent deployments, including an acoustic resonator, a side-scan sonar, underwater cameras, wide-angle cameras, among others [92,93].

Another example is the SWIFTv3 (Surface Wave Instrument Float with Tracking version 3) buoy [101,102] that includes Doppler velocimeters along with motion packages and surface cameras, making it suitable to perform simultaneous measurements of whitecapping coverage and turbulence underneath breaking waves [103,104]. The buoy dimensions are 1.25 m draft, 1 m mast, and a diameter of 0.35 m.

Furthermore, large buoys such as ASIS, SWIFT, or NOC offer the advantage of directly measuring wind speed, wind direction, and momentum flux. This feature potentially makes these types of buoys highly suitable for turbulence quantification for offshore-wind projects. However, the platform motion can impact the accuracy of the measurements and should be accounted for in data analysis and interpretation [105].

On the other hand, smaller hull sizes are advantageous in terms of portability and ease of deployment. They can be quickly transported and launched, making them suitable for short-term campaigns or areas with limited access. Smaller buoys are also generally more cost-effective and require less maintenance. However, they may be more susceptible to bio-fouling which can add mass and potentially lead to buoy sinking. Moored small buoys are well-suited for long-term operational wave monitoring in near-shore regions at relatively low costs. Moreover, when these buoys are set to drift, they prove to be valuable tools for studying ocean currents in remote areas [106]. However, these buoys are limited due to their restricted payload capacity. Ultimately, the selection of buoy hull size and shape depends on the specific deployment requirements, environmental conditions, budget constraints, and desired outcomes of the monitoring campaign.

### 3.4. Comparison of wave measurements

Numerous inter-comparisons of various types of buoys have been conducted under a diverse range of conditions over the years. It is common practice to use Datawell DWR or NDBC 3 m discus buoy data as a benchmark. Herbers et al. [86] compared different GPS-based technologies with traditional heave-pitch-roll buoys, showing strikingly good agreement, even in relatively calm conditions with orbital wave displacements of less than 1 m. However, Veras Guimarães et al. [99] observed a low determination coefficient ($R^2$ = 0.28) and a relatively high normalised root mean squared error (NRMSE = 21.8%) in the peak period when comparing a SKIB buoy with a Datawell DWR. These differences were attributed to the method of estimation of the peak period, which heavily depends on the spectral resolution [67]. Inaccurate estimates of the peak wave period may lead to erroneous calculations of wave energy power, resulting in suboptimal WECs designs; as well as inaccuracies in predicting wave-induced loads, potentially incurring in higher costs.

Lancaster et al. [35] found that a GPS buoy underestimated the wave energy near the spectral peak (around 0.05–0.08 Hz) in comparison to a bottom-mounted ADCP. Similarly, Raghukumar et al. [87] compared measurements of a GPS buoy with a standard Datawell DWR. They found that their GPS-derived significant wave height was within a range of 10% when compared to the Datawell “control” values and the peak wave period within 5% of Datawell measurements. Additionally, the mean wave directions showed a difference of approximately 6°, while the directional spread differences were around 1°. Concerning wave staffs, Drennan et al. [89] and Graber et al. [68] reported that ASIS directional wave spectra are in excellent agreement with those obtained by a NDBC 3 m discus buoy up to 0.35 Hz. Furthermore, the spectral data from ASIS buoys remain valid over a frequency range extended by an order of magnitude.

In summary, both IMU and GPS buoys exhibit similar behaviour, demonstrating their compatibility for wave measurements. The results obtained from the majority of studies consistently show that wave parameters, including significant wave height and wave direction, are in good agreement regardless of the measuring systems used. Some variability is still observed in the peak period data, which may be attributed to differences in data processing rather than the acquisition systems. As suggested by de Vries et al. [71], GPS buoys may be better suited for long waves (periods of up to 100 seconds) than IMU buoys. This kind of waves might include open-ocean swell, resonances in harbours or channels, and potentially even seismic activity. Wave staff systems have demonstrated superior performance, as they combine multiple sensors, providing higher spectral resolution and higher directional resolving power. These results underscore the importance of carefully selecting buoy technologies and data processing strategies, in particular for the estimation of the peak period, to optimise wave characterisation and improve MRE device design, operation, and reliability.

### 3.5. Other applications

Wave buoys can support broader environmental monitoring and ecological applications by incorporating water quality instruments, such as temperature, salinity, dissolved oxygen, pH, turbidity, nutrient concentrations (e.g., nitrate or phosphate), among others [107, 108]. Such data can inform habitat assessments, pollution tracking, and local ecosystem monitoring, extending the buoy utility beyond conventional wave measurements. A key challenge is the integration of diverse sensors for real-time data transmission. Initiatives such as Bristolmouth [109], propose open hardware standards to unify data acquisition across multiple instruments, facilitating environmental impact assessments in MRE contexts.
## 3.6. Recent developments in wave buoy technology

Recent advancements in electronics have facilitated the miniaturisation of MEMS-based IMUs and GPS antennas, enabling the affordable development of compact wave buoys [74,75,99]. Reducing buoy hull size enhances the sensitivity to short waves and increases the frequency resolution [76]. These innovations can make it feasible to conduct extensive deployments, covering vast areas of the ocean at inexpensive costs. Moreover, deploying a spatially distributed array of wave buoys has proven to be an effective approach for accurately characterising directional wave measurements [36].

GPS-based technology has been integrated in commercially available solutions, such as the recently introduced Spotter buoys [35, 87]. This marks a significant advancement in the wider adoption of cost-effective wave measurements while maintaining a high level of accuracy. Furthermore, systems that incorporate both IMU and GPS sensors have demonstrated excellent performance and have shown to be especially well-suited for investigations involving wave-current interactions [99,110], upper-ocean turbulence [101,102] and wave breaking dynamics [91–93].

Finally, open source developments in wave buoy technologies have emerged as a remarkable effort in recent years. Open source platforms, software, and hardware enable the sharing of designs, code, and data, encouraging collaboration and empowering researchers and developers in the field. Innovative solutions, such as the OpenMetBuoy [85], has led to the creation of “an easy to build, affordable, customisable, open source instrument for oceanographic measurements — that you can build yourself or buy commercially”. By actively promoting collaboration and knowledge-sharing, open source initiatives have the potential to revolutionise the field of wave measurements and contribute to a more data-rich and inclusive monitoring framework. Continued efforts in further developing, deploying, comparing, and validating open source technologies will significantly enhance their adoption and impact in the scientific community, enabling wider access to accurate and affordable wave data.

## 4. Acoustic Doppler current profilers

Acoustic Doppler current profilers (ADCPs) have revolutionised the field of physical oceanography by providing detailed measurements of ocean currents [54,111–113]. ADCPs have a wide range of applications, including the observation of large-scale currents in the inner ocean and the measurement of near-surface small-scale turbulence [114–116].

While ADCPs have been primarily designed to measure ocean current velocities, one of their most important applications is the measurement of ocean waves [48,54,117]. This capability extends not only to coastal waters [98], but also to the deep ocean at depths up to 1400 m [118].

In this section, the operating principle and processing techniques of ADCPs are discussed, providing an understanding of how these instruments work. The trade-off between range, resolution, and random noise in ADCP measurements is analysed, exploring the limitations and considerations associated with each. Additionally, this section examines the influence of ADCP geometry, the number of beams and their arrangement, on the accuracy of velocity and turbulence measurements, discussing the advantages and limitations of different beam configurations. Furthermore, a discussion of the various methods to measure ocean waves using ADCPs, particularly the estimation of the directional wave spectrum is presented. Finally, an insight in recent developments and challenges in ADCP technology is offered. Overall, this section aims to provide a comprehensive overview of the use of ADCPs for measuring ocean velocities, turbulence and waves.

## 4.1. Principles of operation

ADCPs use the Doppler principle to estimate the speed and direction of the water currents at multiple cells within a vertical profile. The instrument typically consists of a set of monostatic transducers that emit an acoustic pulse which is reflected by the particles suspended in the water along the path of the acoustic beam [119]. The term monostatic refers to the configuration in which the same transducer both transmits and receives the acoustic signal. If those particles in the water are moving with the ocean currents, then the reflected signal is listened back by the transducer with a shifted frequency. This frequency shift can be interpreted as a change in phase of the acoustic signal over time, resulting from differences in the travel time of the pulse. If the particles are moving away from the transducer, it will take longer for the signal to be received, while if they are moving closer, it will take less time. The Doppler radial velocity along the acoustic beam $u_r$ is then computed as:

$$u_r(t) = \frac{\varphi(t)c_s}{4\pi f_0 \Delta t}; \quad \phi \in [-\pi, \pi], \qquad (17)$$

where $\varphi(t)$ is the phase of the acoustic signal, $c_s$ is the speed of sound in the sea water, which is around 1500 m/s, $f_0$ is the frequency of the transmitted signal, and $\Delta t$ is the length of the acoustic pulse. The vertical cells are defined in terms of the time lag required by the acoustic pulse to cover a specific depth range. Typically, an acoustic pulse will take approximately one millisecond to cover a depth range of 1.5 m. The phase is generally determined using the auto-correlation function of the received acoustic signal [120,121]. The frequency of the transmitted signal can vary from around 50 kHz to more than 2 MHz. Low-frequency ADCPs are typically used in open ocean applications where long profiling range is required. On the other hand, high-frequency ADCPs are useful in shallow water or boundary layer applications where there is a need for precise vertical resolution and the ability to capture short temporal scales [122].

One consequence of estimating the Doppler velocity using Eq. (17) is the potential loss of uniqueness when the signal phase extends beyond the range of $[-\pi, \pi]$. This introduces ambiguity in velocity estimation, commonly known as the phase wrapping limitation [111]. The maximum velocity that can be measured without ambiguity is $u_a = \pm c_s/4f_0\Delta t$, i.e. when $\varphi(t) = \pm\pi$. For instance, a typical 500 kHz ADCP with a pulse duration of 800 µs, would have an unambiguous velocity of the order of 1 m/s. This limitation poses a significant challenge, however, various techniques have been proposed and applied to mitigate this issue [121,123,124].

## 4.2. Processing techniques

According to the characteristics of the transmitted signal, the processing technique or operation mode of the ADCPs can be classified into narrowband, pulse-coherent and broadband. The various processing techniques have implications for their performance, such as maximum measurable velocity, profiling range, uncertainty, power consumption, among others. These factors are summarised in Table 3 and are further examined and discussed in the subsequent paragraphs.

### Narrowband

In narrowband or continuous-wave systems (also referred to as pulse-to-pulse incoherent as successive pulses are uncorrelated), the Doppler velocity is estimated from a single continuous monochromatic pulse of a fixed duration. The range cells are defined by isolating echoes within specific time intervals. The time interval width usually matches the transmitted pulse length. The estimation of the Doppler shift involves identifying the spectral peak of the returning signal coming from a specific range cell [122]. This technique allows long range profiling at the expense of low vertical resolution [125].
<table><caption>**Table 3**
Summary of the main features of ADCP velocity processing techniques.</caption><thead><tr><th rowspan="2">Feature</th><th colspan="3">ADCP processing technique</th></tr><tr><th>Narrowband</th><th>Pulse-coherent</th><th>Broadband</th></tr></thead><tbody><tr><td>Characteristic pulse</td><td>Single continuous pulse</td><td>Pair of separated short pulses</td><td>Sequence of coded short pulses</td></tr><tr><td>Typical application</td><td>Deep ocean currents</td><td>Near-surface or bottom, turbulence, micro-scale</td><td>Versatile, Ocean waves</td></tr><tr><td>Significant limitation</td><td>Low resolution</td><td>Limited profiling range</td><td>Ambiguous velocity</td></tr><tr><td>Maximum Velocity</td><td>Unrestricted</td><td>Limited</td><td>Flexible</td></tr><tr><td>Precision</td><td>Low precision and long averaging times</td><td>Highest precision</td><td>Acceptable precision in most cases</td></tr><tr><td>Profiling Range</td><td>Long: O{100 m}</td><td>Short: O{10 m}</td><td>Medium</td></tr><tr><td>Vertical resolution</td><td>Low: O{1 m}</td><td>Very high: O{1 mm}</td><td>High: O{1 cm}</td></tr></tbody></table>

### Pulse-coherent

On the other hand, pulse-coherent systems transmit a pair of short single-pulse pings separated by a time lag [126]. The first pulse is received and is allowed to fade before the second one is transmitted. Compared with the narrowband system, the pulse-coherent method allows a higher vertical resolution but is limited to short profiling ranges [111,123]. This technology has been useful in single-point Doppler velocimeters and high-resolution profiles of turbulence and micro-structure [114,124].

### Broadband

Broadband systems – introduced by Teledyne RD Instruments in the early 90’s – alike the pulse-coherent technique, transmit a sequence of short-pulse pings, but instead of being separated in time, the pulses are modulated through a pseudo-random repeated code [122] or a chirp-like signal modulation [123,127]. This modulation is intended to exhibit a strong correlation at a specific measurement lag and a minimal correlation at other lags [128]. Broadband technology combines the advantages of the other two techniques, allowing long-range profiling and high vertical resolution [122]. Most of the modern commercially available ADCPs allow the user to select between the different modes of operation according to their specific requirements [123]. Both broadband and pulse-coherent systems has been successfully implemented to measure the surface elevation and orbital velocities caused by waves with high resolution and at high sampling rates [98,114,129].

### 4.3. Range, resolution and random noise trade-off

Generally speaking, ADCP measurements strongly depend on three variables: profiling range, vertical resolution and standard error. ADCP operation frequency determines the profile range: the lower the operation frequency the further the acoustic signal can travel but the lower the vertical resolution.

The standard error strongly depends on the processing technique: in narrowband systems, the standard deviation of the velocity estimation is inversely proportional to the product of the operation frequency and the cell size [119]. This implies that for a fixed operation frequency, a long averaging time is required to control the velocity error [130]. Due to the discrete nature of the pulses, in pulse-coherent systems, the so-called range-velocity ambiguity $u_{max}R_{max} = \pm \lambda_0 c_s / 8$, where $\lambda_0 = c_s / f_0$ is the operation wavelength, $u_{max}$ is the maximum measurable velocity and $R_{max}$ is the maximum depth range, arises as the most important trade-off in this technique, as a consequence, pulse-coherent systems are limited to relatively short depth ranges, on the order of a few tens of metres [122].

For broadband systems, the standard error is more complicated due to the amount of parameters involved. According to Teledyne RD Instruments [119], it is proportional to the ambiguity velocity $u_q$. The most important achievements of the broadband and pulse-coherent technologies over the narrowband are an increase in the vertical resolution, a significant reduction of the velocity variance and a reduction of the averaging times required to control the velocity error [122].

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Illustration of the typical ADCP geometry, including the convention following for the variables naming and the labelling and position of the acoustic transducers.</figcaption></figure>

### 4.4. ADCP geometry and velocity estimation

The typical layout of an ADCP is known as “Janus” configuration. Fig. 4 shows a schematic illustration of this geometry. The convention followed for labelling the transducers may vary among the different manufacturers [119,131]. The illustration shows a coordinate system referenced to the instrument itself, with the origin positioned at the ADCP central transducer. In this particular scheme, the x, y, and z axes in instrument coordinates are aligned with the starboard, forward, and upward directions of the ADCP, respectively. The transformation from instrument coordinates to Earth-referenced coordinates involves the use of the standard three-dimensional rotation matrix. For this purpose, ADCP inclination and orientation respect to north are required. Modern ADCP are equipped with IMU (Inertial Measurements Units) systems to measure their inclination and orientation.

In the “Janus” layout, the beams are inclined an angle $\beta$ between 20° and 30° relative to the vertical. This inclination enables the estimation of the three-dimensional velocity profile relative to the Earth’s frame or reference. By convention, the along-beam velocity is considered positive when particles move away from the transducer and negative when they move towards it. According to the layout presented in Fig. 4 and assuming that the ADCP is placed on a flat bottom with no tilt, the geometrical relationship between the radial velocity $b_n(z, t)$ along the beam n and the instantaneous velocity field $u(x, t)$, with Cartesian components in $u(x, y, z, t)$, $v(x, y, z, t)$ and $w(x, y, z, t)$, can be written as:

$$ 
\begin{aligned}
b_1(z, t) &= u(\delta, 0, z, t) \sin \beta + w(\delta, 0, z, t) \cos \beta \\
b_2(z, t) &= -u(-\delta, 0, z, t) \sin \beta + w(-\delta, 0, z, t) \cos \beta \\
b_3(z, t) &= v(0, \delta, z, t) \sin \beta + w(0, -\delta, z, t) \cos \beta \\
b_4(z, t) &= -v(0, -\delta, z, t) \sin \beta + w(0, \delta, z, t) \cos \beta \\
b_5(z, t) &= w(0, 0, z, t)
\end{aligned} 
\quad (18) $$

where $\delta$ is the horizontal distance from the central beam and $\beta$ is the beam inclination respect the vertical. This set of equations can be
generalised for a generic ADCP layout:

$$b_n(x_n, z, t) = u(x_n, z, t) \cos \alpha_n \sin \beta + v(x_n, z, t) \sin \alpha_n \sin \beta + w(x_n, z, t) \cos \beta \quad (19)$$

where $\alpha_n$ is the orientation of the transducer n in the xy-plane and $x_n$ represents the position $(x_n, y_n)$ of the cell in the horizontal plane, corresponding to the beam n, located at a depth z. Assuming that the velocity field is homogeneous with the ADCP footprint allows for the estimation of the earth-referenced eastward, northward and upward velocity components as:

$$u(z, t) = \frac{b_1(z, t) - b_2(z, t)}{2 \sin \beta} \quad (20)$$

$$v(z, t) = \frac{b_3(z, t) - b_4(z, t)}{2 \sin \beta}, \quad (21)$$

and

$$w(z, t) = \frac{\sum_{n=1}^{4} b_n(z, t)}{4 \cos \beta}, \quad (22)$$

respectively [49,69,111,115].

Apart from measuring the mean velocity profile, ADCPs can also provide measurements of turbulent kinetic energy (TKE) and Reynolds stresses [111,116,124,132]. This information is useful to quantify energy dissipation, mixing and transport processes during wave breaking. To derive turbulence parameters, the along-beam velocities are decomposed into a mean flow and a fluctuating part, as $b = \bar{b} + b'$.
Then, replacing in Eq. (19) and assuming that the turbulence field is horizontally homogeneous, the variance is calculated by squaring and time-averaging over the along-beam velocity. Suitable summations are performed to obtain the following two components of the Reynolds stress:

$$-u'w' = - \frac{\overline{b_1'^2} - \overline{b_2'^2}}{2 \sin 2\beta} \quad (23)$$

and

$$-v'w' = - \frac{\overline{b_3'^2} - \overline{b_4'^2}}{2 \sin 2\beta}, \quad (24)$$

respectively. This methodology can be extended to be written in terms of the co-spectrum, and further considering the contribution of wave-induced orbital velocities in the Reynolds stresses [133,134].

ADCPs traditionally had 3 or 4 slanted-beams, but the addition of a fifth vertical beam allowed for direct measurement of vertical velocity. This five-beam configuration improves the ADCP's ability to directly measure TKE [111]. Nevertheless, Togneri et al. [135] suggested that using the vertical beam does not significantly enhance the estimation of TKE density under low energetic wave conditions. They recommend a five-beam configuration for sites with high wave activity and a particular interest in measuring turbulence near the surface. In their laboratory study, Nystrom et al. [136] evaluated the performance of two pulse-to-pulse coherent ADCPs (a three-beam configuration and a four-beam configuration), with a high-resolution acoustic Doppler velocimeter (ADV). They found that four-beam ADCPs underestimated TKE up to a factor of four. These findings evidence the importance of selecting an appropriate instrument configuration for ADCP deployments and their impact on turbulence measurements.

## 4.5. Wave measurements

### Non-directional wave parameters

ADCPs offer multiple methods for measuring ocean waves, including wave-induced pressure, orbital velocities or surface elevation. Pressure gauges – which are commonly included in modern ADCPs – can measure the dynamic pressure fluctuations caused by ocean waves. As wave-induced pressure attenuates with depth, the conversion from

pressure to surface elevation is challenging, particularly for short wave lengths. Various methods have been proposed, including transfer functions based on linear wave theory [137], non-linear weakly dispersive [138], or non-linear fully dispersive theory [139]. This latter, however, relies on the knowledge of dominant wavenumber spectra, which is rarely known in typical single-point field deployments. In general, the conversion from pressure to surface elevation can be achieved with certain limitations, as it depends on multiple factors, including the relative water depth, sensor sensitivity, data availability, and incident wave conditions. For many practical applications, particularly single-sensors at intermediate to deep water, linear transformation is still the most straightforward alternative.

In orthogonal beams configurations, an estimate of the surface elevation spectrum can be obtained by summing the spectra of the along-beam velocities [49,69,118]. This estimate also considers a linear transformation in a similar way as in pressure-based estimates, but with the advantage of using information of cells near the surface where the attenuation is relatively mild. Nevertheless, the contribution of noise in the velocity measurements may negatively affect the estimates of wave parameters. Alternatively, the incorporation of a fifth vertical beam in ADCPs enabled direct measurements of wave surface elevation using a technique known as acoustic surface tracking (AST). This approach exploits the fact that the maximum backscatter intensity generally occurs at the sea surface [129,140–142]. This is only applicable to bottom-mounted upward-looking ADCPs.

Acoustic-based surface elevation measurements present a more advantageous option when compared to pressure-based detection, as no corrections are required. However AST might be inaccurate in cases when the acoustic return at the surface is dominated by the vertical side-lobe reflection [143]. This primarily occurs during very low wind speeds, when the sea surface is smooth and the reflection is mostly specular [48]. The estimation of Doppler velocities in the upper portion can also be affected by side-lobe contamination. The contaminated range, proportional to the water depth, is $1 - \cos \beta$, which means that for a typical beam angle $\beta = 25^\circ$, the last 9% of the velocity data may be contaminated. This contamination can be extended even further considering cell size [143].

These approaches offer the ability to infer non-directional wave parameters, such as wave height, period, steepness and others, based on time series and spectral analysis. Multiple studies have performed inter-comparison of these different approaches, as well as with other instruments. For instance, Bouferrouk et al. [117] compared several wave parameters obtained by 5-beams ADCP deployed at 40 m in the UK Wave Hub facilities. The comparison was made with four co-located directional wave buoys. Their finding indicate that wave height, mean energy period, and spectral bandwidth, derived from the vertical beam, are in good agreement with estimates from the buoys, exhibiting minimal discrepancies of only 2–7%, with the exception of the peak period. Wood et al. [118] report that the agreement between AST and summed beam-velocity spectra is better than 3%, whereas the agreement between spectra derived from vertical velocity and the four along-beam velocities is 10%. Lancaster et al. [35] report mean values of the significant wave height and peak wave period agree within 3%, in comparison with a Spotter buoy. They further compared their measurements with pressure gauges. They found significant deviations between the pressure-derived significant wave height and peak period, in comparison to those estimated by the ADCP and the Spotter buoy. The differences were attributed to linear conversion from pressure to surface elevation, affecting frequencies above 0.25 Hz. These findings further confirm the ability of the vertical beam to improve estimates of non-directional wave parameters. However, additional comparisons of the multiple alternatives for extracting these parameters from ADCPs are necessary.
## Directional wave spectrum

Two different approaches can be adopted in order to estimate the directional wave spectrum using ADCPs. The first consists in using three wave characteristics: typically the two horizontal components of the wave-induced orbital velocities ($u$ and $v$) and a non-directional quantity, that can include pressure [144], vertical velocity [118] or echo-based surface elevation [48]. This approach is known as is “triplet” or “triple-point” as it composed by three measurements at one single location. The second approach, on the other hand, considers that the ADCP beam geometry constitutes an array of wave measurements [70,141]. In this approach, the horizontal separation of the along-beam measurements allows for the estimation of the wave direction, considering the lags or phase differences associated with waves passing through each element of the array.

Both triplet and array processing techniques may present some limitations and restrictions. Strong et al. [145] presents a comprehensive comparison of these approaches. In general, the main limitation is inherited from the transformation from pressure or velocity to surface elevation as discussed earlier. The triplet approach requires in addition the assumption of homogeneity of the velocity field to obtain the Earth-referenced velocities. The array processing, on the other hand, may restrict the range of resolvable wavelengths depending on the beam aperture. For very long waves, the phase difference between ADCP beams is close to zero, implying that wave direction cannot be accurately determined. On the other hand, large separations between beams may be an issue for short waves. As an example, consider an upward-looking ADCP with a beam inclination of $\beta = 25^\circ$ and mounted at a depth of 50 m. The near-surface beam aperture for this configuration, assuming a central beam, is approximately $\delta = 21$ m. This implies that the smallest measurable waves have a wavelength of $\lambda = 10.5$ m, which corresponds to a cut-off frequency of 0.4, Hz. However, sub-surface cells can be used to increase the maximum resolvable wavelength at expense of signal attenuation.

In summary, the versatility of ADCPs offers multiple possibilities for estimating the directional wave spectrum. The triplet and array processing techniques provide valuable information but also present constraints that need to be addressed for accurate results. Exploring alternative approaches like correlations of consecutive cells along the same beam could offer significant improvements. Additionally, signal fusion algorithms, like complementary filters or Kalman filters, which combine the most suitable approaches for specific conditions, could provide an enhanced representation of the directional wave spectrum. Therefore, continued investigation and innovation in ADCP-based wave measurement techniques are essential to unlock their full potential.

## 4.6. Other applications

An interesting application of ADCPs includes estimating zooplankton distributions and monitoring fish activity. ADCP backscatter intensity is used to infer the vertical distribution and abundance of zooplankton in the water column, which is valuable for understanding ecosystem dynamics, seasonal migration, and nutrient cycling in the marine environment [146,147]. Moreover, the development of algorithms to detect discrete targets based on the backscatter signal, may enable ADCPs to become an essential tool in biological monitoring [148], thereby aiding in the assessment of the environmental impact of MRE projects. Another alternative application involves the estimation of bubble plumes in the upper ocean. These characteristics include bubble plumes vertical extension, lifetimes and events periodicity. This information is essential for characterising wave breaking probability and other processes, such as, bubble-mediated gas exchange [29,92,93,149–152]. These alternative applications demonstrate the adaptability of ADCP beyond its conventional use in measuring ocean currents and waves.

## 4.7. Recent developments and challenges

In recent years, an interest in increasing the number of beams has been raised, particularly with the aim of improving the estimation of turbulence parameters. Vermeulen et al. [132] presented a technique to measure the Reynolds stresses using a two coupled four-beam ADCPs system. They observed that in eight-beam systems, it is possible to estimate the Reynolds stress tensor completely, without requiring any additional assumptions, other than the homogeneity of the turbulence field between the beams span. More recently, Droniou et al. [153] analysed several configurations of ADCPs to evaluate the ability to measure waves and turbulence. They concluded that the seven-beam configuration (six slanted-beam plus one vertical) presents the best compromise and indicated that the vertical beam is important to measure directly the vertical component of the velocity and get a robust measurement of the surface elevation. In addition, the three-beam ADCP velocities were as much as 17% lower than velocities measured with the ADV and both TKE and Reynolds stresses are underestimated by the three-beam ADCP.

An interesting new design was presented by Hay et al. [127]. They described a modular Doppler sonar measuring system, called “Vectron”, composed by several single-beam ADCPs separated by 6 m, converging on a focal point. This system was developed to obtain accurate measurements of turbulence, particularly in sites where flow speeds are as high as 5 m/s. A similar approach was followed by Harding et al. [154]. The concept of a converging ADCP configuration, known as C-ADP, has been introduced as a significant improvement for turbulence measurements, however future field validations of these systems is required.

Striking new developments in ADCP technology in recent years have further enhanced their capabilities and expanded their versatility. One significant advancement is the broadband Acoustic Doppler Dual Current Profilers (AD2CP) system introduced by Nortek AS in their Signature series (Patent US 7,911,880 B2, March 22, 2011). This platform incorporates alternating acoustic sampling with varying transmit pulse and signal processing configurations, allowing for quasi-simultaneous measurements within the same water volume using two “virtual ADCPs” [123]. In addition, Rowe Technologies coined the term “3rd generation ADCPs” (ADCP3) to systems capable of operating at two or three different frequencies simultaneously (Patent US 10,698,107 B2, June 30, 2020 and US 2019/0257930 A1, August 22, 2019). While these systems have the potential to enhance the versatility of ADCPs, additional research is necessary to fully explore their applications, especially in the context of the wave field characterisations for MRE projects.

# 5. Hydrophones

Hydrophones are simply underwater microphones. They are designed to detect and measure sound waves in aquatic environments. Although hydrophones are widely employed in diverse fields of science, engineering, and industry, including for example, marine biology, navigation, underwater construction, and military applications, they can also serve the purpose of measuring specific attributes of ocean waves. This section presents a brief overview of hydrophones, including their principles of operation and essential parameters, focusing primarily on their applications for measuring acoustic characteristics of ocean waves.

## 5.1. Principles of operation and underwater acoustics

Underwater acoustics is the study of sound propagation in water environments, such as rivers, lakes or the ocean. Acoustic waves propagate in the water at a speed that depends on temperature, salinity, and pressure. The average sound speed in the water is around 1500 m/s, more than four times the sound speed in the air. The key principles
of underwater acoustics involve the generation, propagation, and detection of sound in water. Sound sources can be natural (e.g., marine mammals, turbulence, bubbles) or anthropogenic (e.g., ships, sonar). In particular, hydrophones are essential for detecting acoustic signals in aquatic environment. A detailed introduction to underwater acoustics, including the description of physical mechanisms, models of sound propagation, acoustic data processing, and typical units, can be found in Kuperman [155]. The content of this sub-section is mostly based on the valuable work presented by Saheban and Kordostami [156], who conducted a thorough review of hydrophones, covering various aspects such as design considerations, physical attributes, operational mechanism, frequency response, and applications.

Traditionally, hydrophones operate based on the piezoelectric principle, where specific materials generate an electric charge in response to mechanical pressure. When an acoustic wave propagates through water and reaches the hydrophone, it induces a slight deformation in the piezoelectric material. This deformation leads to a measurable change in voltage, which can be recorded and interpreted as a sound. The advancement of MEMS technology has significantly enhanced the progression of hydrophone development, similar to the discussed case of IMUs (Section 3.1). These systems offer several advantages, including small size, high sensitivity, desired low-frequency response, good consistency, light weight, low power consumption, high reliability, proper robustness, mass production capability, and affordable pricing. These attributes make them a viable and attractive alternative to the wide range of hydrophones available today.

Frequency response, sensitivity, and maximum operating depth are crucial specifications that must be considered when evaluating hydrophones. The audible spectrum for humans usually extends from 20 Hz to 20 kHz. In contrast, the frequency band for hydrophone applications covers a much broader spectrum, ranging from roughly 1 Hz to several MHz. Similarly, healthy humans can hear sound amplitudes approximately between 0 dB (total quietness) to 120 dB (pain levels) SPL.<fnref n="1" /> As a reference, commercial hydrophones are highly sensitive, reaching sensitivities between –197.7 dB and –155.6 dB re 1V/µPa. Finally, hydrophones can be deployed at depths the order of thousands of metres. These deployments at great depths typically require pressure-resistant housings to protect the electronics from the high pressures at those depths. In summary, this highlights the capability of hydrophones for underwater acoustic measurements, which extends beyond the scope of human hearing.

## 5.2. Applications to ocean waves

Interestingly, underwater noise captured by hydrophones may be considered a negative factor in certain applications, particularly when it obscures and masks sounds emitted by specific targets such as fish, mammals, or ships. However, this very noise is of significant importance in studying processes involving wind and waves. Subsequently, this section explore two specific cases where hydrophone recordings prove valuable in analysing ocean waves.

### Bubbles produced by breaking waves

Underwater acoustic noise is predominantly influenced by wind-induced processes, making them the main contributor to ambient noise across a broad spectrum of frequencies, ranging from 400 Hz to 50 kHz [157–159]. In particular, bubble plumes produced by wind-generated breaking waves contribute significantly to the underwater noise levels [92,93,160]. The sound produced by these bubbles can be attributed to individual bubble oscillations [161,162] and collective oscillations of bubble plumes [163]. The frequency of sound generated by

individual bubbles is determined by their radius [164], and non-linear shape oscillations of bubbles can resonate with the radial oscillation, producing peaks in the noise spectrum [161]. Moreover, laboratory and field measurements support the notion that wind-generated noise is primarily caused by bubbles produced by breaking waves [162,165].

The analysis of breaking waves through the noise generated by their bubbles is commonly referred to as passive acoustic measurements. This subject have been extensively studied by numerous authors.

Farmer and Vagle [166] employed a hydrophone to obtain the sound generated by breaking surface gravity waves in Georgia Strait, British Columbia. Their results provide valuable information about the relationship between the acoustic signal of breakers and their mean spacing. Ding and Farmer [167] improved this technique, incorporating a directional array of hydrophones and a method to track individual breaking events. Loewen and Melville [168] proposed a theoretical model that establishes a relationship between the bubble size distribution and the acoustic spectrum. This model allows the estimation of the bubble size distribution without the need to explicitly identify individual bubble pulses in the acoustic time series.

More recently, Zou et al. [169] analysed the frequencies of the sound waves generated by newly formed bubbles and established a linear relationship between the mean bubble radius and the breaking severity. This relationship provides a means to quantify the severity of wave breaking based on the size of the bubbles produced.

In summary, the most important outcomes of passive acoustics determination of breaking waves include the identification of breaking events and their association with wave parameters [166], the quantification of the amount of energy dissipated during the breaking process [169,170], and the estimation of the distribution of bubbles sizes [169,171].

### Underwater noise and microseisms

An interesting yet relatively uncommon application of hydrophones is the study of microseisms, the origin of which has been a fascinating and intriguing topic for many years [172,173]. Almost eight decades ago, Longuet-Higgins [174] proposed a theory that explains the generation of microseisms by the acoustic radiation of a second-order pressure fluctuation at the bottom of the ocean. This pressure fluctuation, which is not attenuated with depth, is caused by waves travelling in opposite directions. Pellet et al. [175] derived expressions for the general case of waves, with nearly equal frequencies, travelling in nearly opposite directions. The frequency of resulting second-order pressure is twice the frequency of the carrier waves.

As the second-order pressure fluctuation is unaffected by depth, it remains detectable at the ocean floor. Farrell and Munk [51,176] employed hydrophones, deployed at 5000 m depth, to measure the acoustic spectra produced by these pressure fluctuations in the frequency band 1–30 Hz. One striking result was the transition from $f^{-7}$ to $f^{-3}$ slope in the acoustic spectra occurring between 4–6 Hz. This was attributed to the transition from the classical Phillips spectrum to ultra-gravity waves band, occurring between 2–3 Hz in the surface waves spectrum. Similarly, Duennebier et al. [52] utilised data collected by a broadband hydrophone positioned 10 m above the ocean floor in water depth of 4720 m at the ALOHA Cabled Observatory, situated 100 km north of Oahu, Hawaii. They conclude that the acoustic spectrum, between 0.4 Hz and 80 Hz, can be explained by the Longuet-Higgins [174] mechanism.

This phenomenon becomes particularly interesting when exploring its connection with the bimodal nature of the directional distribution of wind-generated waves [177,178]. In these scenarios, a significant proportion of waves, shorter than the dominant scale, propagate in nearly opposite directions, producing a distinct signal at the ocean floor. In conclusion, the use of hydrophones, even several kilometres away from the sea surface, can offer valuable insights into the directional spreading of surface gravity waves.

<footnote n="1">¹ The units of dB SPL are defined as $20 \log_{10}(p/p_{\text{ref}})$ where p is the measured sound pressure level and $p_{\text{ref}}$ is a reference value, which is typically 20 µPa. It means the 0 dB and 120 dB are in fact equivalent, in terms of sound pressure levels, to 20 µPa and 20 Pa, respectively.</footnote>
### 5.3. Limitations and challenges

There are limitations and challenges associated with the use of hydrophones, particularly for measuring ocean wave acoustic characteristics. One key challenge is the difficulty in data processing. Acoustic information from individual bubbles can be complicated to obtain as it may be masked by moving parts of the mooring line or affected by undesired biological activity. Improved signal processing algorithms could be a powerful tool for that purpose. By training models on large datasets of known bubble signals, it could be possible to develop automated techniques for accurately detecting and quantifying individual bubble parameters based on their acoustic signatures. Although this approach can yield valuable insights, further laboratory and field experiments are required to refine and validate these methods.

Regarding their deployments, hydrophones can be deployed either cabled or battery-powered. While cabled deployments allow for data transmission and virtually eliminate power supply issues, they can be complicated at greater depths. On the other hand, battery-powered hydrophones can be deployed at greater depths, but their autonomy in terms of power and storage capacity is limited. Furthermore, data from battery-powered hydrophones can only be accessed upon recovery, which may lead to unexpected results. Additionally, for ocean wave research purposes, it is essential to obtain simultaneous wave measurements along with the hydrophone data, as this enables a comprehensive understanding of the relationship between ocean waves and the corresponding acoustic signals recorded by the hydrophone.

Lastly, it is important to note that hydrophone manufacturers typically provide a range of products specifically tailored to meet different research purposes. For instance, they may offer high-frequency hydrophones that are ideal for capturing detailed acoustic information, or very low-frequency hydrophones that excel at detecting deep-sea ambient noise. Consequently, researchers must possess a thorough understanding of hydrophone specifications and characteristics before making a selection. This requires considering factors such as sensitivity, frequency response, and dynamic range, to ensure that the chosen hydrophone aligns with the specific research requirements and objectives.

## 6. Other wave measurement techniques

### 6.1. On-shore or vessel-mounted remote sensing

#### Stereo-imaging and optical systems

The use of stereographic imaging to estimate the directional wave spectrum dates back to the early 20th century [179]. Since then, advances in digital imaging over the past few decades have transformed this method into a practical tool for wave measurements. Stereo-imaging creates a depth perception using two synchronised cameras placed at a fixed distance apart. The pair of stereo-images undergoes a series of processing steps, including camera calibration, geometrical alignment, lens distortion correction, and image rectification [180]. Corresponding features between sequences of stereo pairs are identified and matched to obtain a dense 3D representation of the observed sea surface. The point cloud is projected and interpolated onto a regular grid, obtaining the reconstruction of the time evolution of the sea surface elevation, $\eta(x, t)$, over a specific area [181]. Various approaches have been explored to improve the accuracy of surface reconstruction but at a higher computational cost [182]. The key advantage of stereo is that it produces the full three-dimensional wave spectrum, $E(k, \omega)$.

In general, the accuracy of stereo measurements depends on factors such as camera calibration, image resolution, sea roughness and luminosity. However, the most determinant factors are the camera separation (baseline), viewing angles and the distance from the sea surface (height). These factors affect the resolutions in both the z (depth)

and x, y (horizontal) dimensions, as well as the field of view. Benetazzo [181] suggests a height to base ratio of 0.1 as an optimal configuration. The trade-off lies in the size of the captured image: larger images are suitable for capturing long swells but may result in lower resolution, while smaller images are well-suited for capturing short waves but require higher resolution. Consequently, stereo-imaging is particularly well-suited for high-resolution analysis of short waves, including the wavenumber spectra [183], and the short-scale directional characteristics [184]. Stereo-imaging has also proven to be instrumental to study waves in extreme conditions [37,38,185]. In addition, researchers have produced high-quality open-access databases of stereo-images to increase accessibility and promote familiarity with these techniques [39].

The use of cameras for obtaining information about the wave field is not exclusive to stereo techniques. Single optical images can also been used for this purpose [186]. Recent advances in artificial intelligence and increased computational power have facilitated the application of machine learning algorithms for the analysis of single or stereo optical images. Specifically, the use of different neural network architectures enables the identification and classification of wave breaking [187, 188]. This not only opens up new possibilities for better understanding wave breaking physics, but also becomes a valuable tool for accurately assessing and predicting the impact loads caused by breaking waves, ultimately preventing structural damage [189].

#### Radar-based wave measurements

Radar systems can be deployed onshore or mounted on vessels, aircraft, or satellites (see further details on satellites in the Section 6.2). Common types include SAR (Synthetic Aperture Radar), HF (High-Frequency) radar, and X-band radar, each operating in different microwave frequency bands. This latter, typically used in marine navigation, has attracted increased attention to measure ocean waves in the recent years [60]. Similar to stereo-imaging, radars are also capable of producing the full three-dimensional wave spectrum, $E(k, \omega)$, through conventional Fourier analysis [190]. In addition, they provide information on wind and surface currents [191].

Radar imaging of the sea surface (sea clutter) involves two primary mechanisms: microwave specular reflection at near-vertical incidence angles (used in satellite and aircraft-mounted altimeters) and Bragg resonance backscatter at grazing incidence angles (used in vessel-mounted and on-shore configurations). X-band radars operate in the frequency range 8–12 GHz (approx. 3 cm wavelength), meaning that it is the gravity-capillary part of the wave spectrum the responsible for the Bragg backscattering. The backscatter signal is further modulated by longer waves, which allows X-band radars to measure their characteristics [190]. However, this modulation is non-linear and hinges on a comprehensive understanding of mechanisms such as shadowing and tilt modulation to accurately transform radar images into physical surface elevation data (the inversion problem) [40,41,192]. The development of coherent X-band radars, which use both amplitude and phase information from the backscatter signal, has represented an advancement for radar technology. They have shown utility in identifying wave shoaling and breaking in shallow waters [42].

Among the limitations affecting radar performance, rainfall is particularly important. Reducing the operation frequency to 2–4 GHz (S-band) has been considered as a potential alternative to mitigate some of the interference caused by rain, compromising, however, the image resolution [193]. The versatility of X-band radars goes beyond the estimation of physical parameters. McCann and Bell [194] demonstrated the capability of these systems in detecting and monitoring marine mammals, which could prove valuable in supporting environmental impact assessments for MRE initiatives.
## 6.2. Satellite-based wave measurements

The advent of satellite-based remote sensing technologies has significantly enhanced our capacity to measure ocean waves on a global scale. A recent literature review by Hauser et al. [44] has synthesised existing knowledge on satellite-based remote sensing of wind, waves, and currents, covering topics ranging from measurement techniques and historic aspects to applications in coastal management and climate research. Notwithstanding the extensive coverage of their review, which serves as a definitive resource for the field, a brief mention of key satellite missions and their potential applications in the context of MRE, specifically in relation to ocean waves, is provided below.

The CFOSAT (China-France Oceanography SATellite) mission stands up as pioneer initiative to obtain the directional wave spectrum from satellites [43,195,196]. Its on-board instrument SWIM (Surface Waves Investigation and Monitoring) includes a Ku-band radar (12–18 GHz) with a rotating antenna, allowing for wave observations at different azimuth, thereby composing the directional wave spectrum. The CFOSAT/SWIM data may play thus a pivotal role for long-term site characterisation in MRE projects. Its main strength is the detailed directional information and wave partitions. However, the mission has limitations such as low measurement frequency due to its orbit cycle of approximately 13 days, limited resolution, and the inherent 180-degree ambiguity in directional wave data.

Multiple altimeter missions, in orbit for several decades, have been fundamental in validating numerical wave models and analysing long-term trends wave parameters [55]. However, they only provide wind speed and significant wave height and do not natively capture wave period or direction [197]. SAR imagery is also a significant asset in the study of ocean waves, with the Sentinel-1, ENVISAT, and TerraSAR-X missions standing out as a notable examples. The directional wave spectrum (with 180-degree ambiguity), and thus the wavelength and wave direction can be derived from SAR images [198–200]. Looking towards the future, innovative concepts like SKIM (Surface Kinematics Multi-scale), that uses Doppler capabilities to measure simultaneous waves and currents [201], promise to enhance our understanding of wave dynamics, and consequently advancing the efficiency and reliability of MRE installations.

## 7. Critical aspects of wave measurements and analysis

### Challenges in capturing non-linearities

Ocean waves are fundamentally non-linear in nature, particularly in shallow waters, under strong currents, or when interacting with structures. Floating structures respond to different non-linear waves features, including second-order low-frequency and high-frequency wave loads, wave slamming, and wave breaking [202].

These non-linearities are typically investigated using coupled wave-structure numerical models. Consideration of non-linear wave loads is typically advised by standard specifications for MRE projects [203]. However, from a measurement perspective, fully capturing these non-linear effects can be challenging due to their complex nature and inherent limitations of wave measuring techniques. A clear manifestation of wave non-linearity is the sharpening of the crests, particularly evident for steep waves [204,205].

There has been ongoing discussion on whether wave buoys are able to capture these extreme crests. [204] noted that surface-following buoys effectively cancel out second-order non-linearity by making Lagrangian measurements. Factors such as buoy inertia, surface sliding, mooring tension, or internal signal processing artifacts can lead to an underestimation of extreme crest heights by buoys [206]. Although these non-linearities are not expected to significantly impact conventional sea-state parameters estimated from wave buoys, further research in this area is indeed necessary.

### Cost considerations and scalability challenges

Field data collection involves costs associated with sensor purchase, technical support, ship operations, deployment, maintenance, and decommissioning. These expenses vary depending on the measurement technology used, with high-precision instruments requiring substantial capital investment and operational costs. In the long-term, operational expenses, especially those associated with power consumption, such as battery replacements, maintenance of solar panels, and real-time data transmission, can become a significant constraint. To address these challenges, wave-powered buoys have been suggested as a means to reduce power-related expenses in locations with significant wave energy potential [207].

From an economic perspective, data from integrated networks generate more benefits than fragmented systems [208]. Extensive buoy networks such as CDIP (Coastal Data Information Program) and NDBC have been instrumental in advancing wave research and supporting MRE projects [209]. However, their spatial coverage is limited, restricting their applicability in emerging MRE regions [210]. Satellite-based measurements provide a complementary approach, enhancing global coverage and reducing costs for end users, as they are often funded through public programs.

Scalability challenges arise in expanding wave observation networks, particularly in real-time transmission, storage and processing of large volumes of data [211,212]. Exploring low-cost alternatives and open hardware standards are essential in mitigating the economic challenges and ensuring scalability [85,87,109]. However, data quality of low-cost sensors must be rigorously assessed against advanced sensors. Finally, the increasing deployment of low-cost sensors raises concerns about environmental impact and long-term sustainability. Addressing marine litter issues and promoting ecological responsibility in sensor designs will be essential as wave measurement networks continue to expand.

### Regulatory and policy challenges

Deploying measurement systems at sea often requires regulatory approval, which varies by jurisdiction and can delay projects [213]. Data transferability between projects can reduce approval timelines [214], but site-specific variations and project requirements must be considered to ensure its applicability in wave measurements. The absence of standard specifications is another key regulatory challenge, especially for wave energy resource assessments [215]. The wide variety of wave sensor types and their principles of operation present obstacles for establishing standard procedures, especially for the calibration, validation and inter-comparison of wave measurements.

Environmental regulations play a vital role in preventing marine litter resulting from measurement instruments and in avoiding disruptions to marine life or navigation. Hubert [216] highlights the juxtaposition between the positive contribution of marine exploration in advancing ocean understanding and the concerns about the adverse impacts of scientific research in the marine environment. For instance, it is important to establish clear protocols and assign responsibilities in the event that a buoy detaches from its mooring, to ensure prompt retrieval and prevent potential hazards. Another important consideration is the possible acoustic impacts generated by ADCPs. Although the acoustic frequencies typically used by commercial ADCPs are generally considered safe for marine life, new developments should aim to further minimise any potential impacts on marine ecosystems. Additionally, proper management of chemicals, such as batteries, is essential for minimising environmental risks.

Cybersecurity aspects should be considered to ensure data integrity and reduce vulnerability in connected observation networks [217,218]. Additionally, factors such as vandalism in regions, where law enforcement is challenging, present significant obstacles for continuous wave monitoring [108,219].
<caption>**Table 4**</caption>

Summary of the wave measurement systems outlined in this study. The comparison is based on their capability, costs, spatial and temporal coverage, and suitability for MRE applications. Two types of buoys have been included to demonstrate the varying characteristics that can result based on their size and design.

<table><thead><tr><th>Sensor</th><th>Capability</th><th>Cost</th><th>Coverage</th><th>Suitability for MRE</th></tr></thead><tbody><tr><td>Small buoy</td><td><strong>Acceptable</strong> – Motion sensors, GPS, real-time data, ideal for bulk wave parameters</td><td><strong>Low</strong> – Affordable, easy deployment, low maintenance</td><td><strong>Point</strong> – Deployments can last for several months, often used in drift mode</td><td><strong>High</strong> – Cost-effective, versatility, ideal for MRE site monitoring</td></tr><tr><td>Large buoy</td><td><strong>High</strong> – Precise wave sensors, good for scientific research</td><td><strong>High</strong> – Expensive upfront costs, deployment, mooring, maintenance</td><td><strong>Point</strong> – Ideal for long-term wave data, used in monitoring networks</td><td><strong>High</strong> – Reliable for feasibility studies, long-term monitoring, scientific research</td></tr><tr><td>ADCP</td><td><strong>High</strong> – Measures waves, currents and turbulence, accuracy depends on sea conditions and setup</td><td><strong>High</strong> – Expensive, power-demanding, complex deployment</td><td><strong>Point</strong> – High-resolution but limited spatial data, vertical profile data</td><td><strong>High</strong> – Critical for subsurface currents, floating MRE systems</td></tr><tr><td>Hydrophone</td><td><strong>Medium</strong> – Measures certain wave features via acoustic signals</td><td><strong>Low</strong> – Affordable, depends on setup</td><td><strong>Point</strong> – Captures sound over an area, help detecting bubbles and breaking waves</td><td><strong>Medium</strong> – Supports environmental studies, not suitable for wave parameters</td></tr><tr><td>Satellite</td><td><strong>Medium</strong> – Accuracy depends on sensor type, affected by noise, complicated transfer functions</td><td><strong>Low</strong> – Generally public funded, data freely available for end users</td><td><strong>Global</strong> – Broad coverage, low temporal and spatial resolution, long-term monitoring</td><td><strong>Medium</strong> – Good for climate studies, lacks fine-scale resolution, useful for model validation</td></tr><tr><td>Radar</td><td><strong>Medium</strong> – Directional wave spectrum, wind, currents, difficulties for wave height</td><td><strong>High</strong> – Expensive setup, maintenance, power demand</td><td><strong>Regional</strong> – Covers coastal areas, limited offshore range but can be used on ships</td><td><strong>High</strong> – Real-time wave mapping, ideal for coastal MRE sites</td></tr><tr><td>Camera</td><td><strong>High</strong> – Full wave spectrum, AI enhances wave classification, limited to daylight</td><td><strong>Low</strong> – Affordable, easy to deploy, short life, computational costs for processing</td><td><strong>Local</strong> – Limited to the field of view of the cameras and deployment conditions</td><td><strong>Medium</strong> – Good for detailed wave features but limited for large-scale monitoring</td></tr></tbody></table>

## Future research directions

Future research should focus on advancing sensor fusion techniques to integrate multiple data sources for improved accuracy and reliability. Developing strategies is crucial to ensure continuous operation in harsh marine environments, particularly through enhanced sensor durability in extreme conditions such as high waves, strong currents, and bio-fouling [16,100]. Machine learning for data analysis can optimise real-time processing, automate quality control, and enhance predictive modelling of wave dynamics [187,188]. Additionally, innovations in new materials and sensing technologies can improve sensor efficiency, longevity, and environmental sustainability, enabling more robust and cost-effective wave measurement solutions [220].

## 8. Summary and conclusions

This study has reviewed different wave measuring techniques that are highly relevant and applicable to MRE initiatives. An exploration of the underlying working principles, advantages, and limitations of various wave sensors, including surface wave buoys, ADCPs, and hydrophones has been presented. These instruments offer convenience and ease of access for researchers and engineers, as they are commercially available and provide valuable data for describing ocean waves. A mention of remote sensing methods such as stereo-imaging, radars and satellite was also included. The information that can be extracted from each sensor and the commonly used analysis techniques to obtain meaningful data has been analysed. Furthermore, this review has highlighted recent technological developments and advancements in each sensor type, focusing on their implications for understanding ocean wave dynamics. The main insights presented in this study are:

* Wave measurements are vital in MRE projects. Examples include using wave data for site selection, performance optimisation, evaluating loads and structural fatigue, and ensuring safety

operations at sea. An accurate wave characterisation, achieved through *in-situ* measurements, in combination with numerical models and remote sensing, is essential for ensuring the efficiency and sustainability of MRE operations.

* Various approaches exist for measuring waves, including *in-situ* point measurements using wave buoys, ADCPs, and hydrophones; and remote sensing techniques using satellites, radars and cameras. All these exploit different wave aspects such as kinematics, acoustics and their interactions with microwaves.

* The adoption of technologies, such as MEMS-based electronics and GPS buoys, has substantially improved the reliability of wave measurements. This technological progress has not only improved sensor performance but has also reduced costs, thereby enabling widespread implementation.

* Recent advancements in the field of ADCPs have enhanced their capabilities for measuring ocean velocities, turbulence, and waves, including the development of broadband ADCPs and the integration with other sensors, thus offering more accurate site characterisation and supporting the design and operation in the MRE sector.

* Hydrophones, known for their cost-effectiveness and versatility, are primarily used in the industry for monitoring underwater noise and marine life detection. They have also shown utility for wave measurements, including the detection of bubbles produced by breaking waves, the study of microseisms and directional wave spreading. This not only supports environmental impact studies but also enhances understanding of wave dynamics, benefiting the MRE community.

* Challenges still exist in the field, including power-efficient deployments for long-term monitoring, better understanding of the directional characteristics of waves, and improved integration of wave measurements with other sensors.

* Remote sensing techniques including stereo-imaging, radars, and satellite offer high spatial resolution for detailed wave analysis, complementing *in-situ* measurements.
Selecting appropriate wave measurement instruments requires balancing accuracy, durability, and operational constraints. As it was discussed, different sensors offer distinct advantages and limitations depending on environmental conditions, deployment duration, and data requirements. Table 4 summarises the main characteristics of the wave sensors presented here, highlighting the trade-offs to assist in the selection of suitable instruments for wave measurements in the context of MRE applications.

Various challenges in existing knowledge have been identified throughout this research, including the difficulties in acquiring accurate and reliable measurements across large spatial scales and extended time periods, the damaging effects of the harsh marine environment on measurement devices, and limitations in measuring technologies. Addressing these challenges is vital for the successful integration and optimisation of MRE technologies, ultimately contributing to the transition towards renewable and sustainable energy sources.

## CRediT authorship contribution statement

**Daniel Peláez-Zapata:** Conceptualization, Investigation, Writing – original draft. **Vikram Pakrashi:** Supervision, Writing – review & editing. **Frédéric Dias:** Supervision, Writing – review & editing, Funding acquisition.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was funded by the European Research Council (ERC) under the EU Horizon 2020 research and innovation program. Grant Agreement No. 833125-HIGHWAVE (DPZ, FD) and by Science Foundation Ireland (SFI) under Marine Renewable Energy Ireland (MaREI), the SFI Research Centre for Energy, Climate and Marine (Grant No. 12/RC/2302). VP also acknowledges NexSys 21/SPP/3756, SEAI RDD/966 FlowDyn, RDD/613 RemoteWind, Interreg Atlantic Area SISdATA and EAPA_00402022. DPZ acknowledges Melissa Jaramillo-Torres for her comments and suggestions. We acknowledge the following people and institutions for the use of buoy images in Fig. 3: Marco Larrañaga and Francisco Ocampo-Torres for the BOMM image; the Datawell image is credited to Rémi Kaupp; the Spotter image is used with permission from Sofar Ocean; the MetBuoy image is credited to Arnaud Disant; and the NOMAD image is in the public domain.

## Data availability

Data will be made available on request.

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