Marion Huchet
email: marion.huchet@dynamocean.com

Aurélien Babarit

Guillaume Ducrozet

Pierre Ferrant

Jean-Christophe Gilloteaux

Eloi Droniou

Experimental assessment of a nonlinear, deterministic sea wave prediction method using instantaneous velocity profiles

Keywords:

List of symbols and abbreviations

Introduction

As opposed to stochastic prediction, which yields only averaged information on the wave spectrum, deterministic sea wave prediction can provide a wave-by-wave description of the sea state's characteristics: surface elevation, kinematics, wave slope, etc. This information, if available in real-time, would greatly benefit a wide range of marine applications, such as dynamic ship positioning, improving the safety of operations, or developing control strategies for wave energy converters or offshore wind turbines.

Despite recent progress in computational capabilities and the development of new oceanographic sensors, realtime deterministic wave prediction remains a challenge and several families of methods are still under investigation. Among them, methods based on Machine Learning are on the rise and show promising results [START_REF] Duan | Phase-resolved wave prediction model for long-crest waves based on machine learning[END_REF][START_REF] Zhang | Phase-resolved real-time ocean wave prediction with quantified uncertainty based on variational Bayesian machine learning[END_REF]. But the currently most advanced approaches for operational purposes use propagation models based on linear wave theory, because of the very short associated computational time. Some solutions are already commercially available [START_REF] Hilmer | Observations of predictive skill for realtime Deterministic Sea Waves from the WaMoS II[END_REF][START_REF] Kusters | FutureWaves™: A real-time Ship Motion Forecasting system employing advanced wave-sensing radar[END_REF][START_REF] Naaijen | Validation of a Deterministic Wave and Ship Motion Prediction System[END_REF]. They offer satisfying results, provided the space and time scales of propagation are short, the prediction horizon is limited and the characteristic wave steepness is small. If these conditions are not met however, nonlinear effects become significant during wave propagation and the prediction quality deteriorates [START_REF] Toffoli | Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory[END_REF][START_REF] Bonnefoy | Time domain simulation of nonlinear water waves using spectral methods[END_REF][START_REF] Zhang | An experimental and numerical study on breather solutions for surface waves in the intermediate water depth[END_REF].

Therefore, despite greater computation times, other approaches use partially nonlinear wave models, up to the third order [START_REF] Trulsen | Spatial Evolution of Water Surface Waves: Numerical Simulation And Experiment of Bichromatic Waves[END_REF][START_REF] Adcock | The nonlinear evolution and approximate scaling of directionally spread wave groups on deep water[END_REF][START_REF] Simanesew | Surface wave predictions in weakly nonlinear directional seas[END_REF], or even fully nonlinear models. Among the latter, the HOS method was found to be particularly promising for nonlinear wave prediction over large space-time domains [START_REF] Wu | Direct simulation and deterministic prediction of largescale nonlinear ocean wave-field[END_REF][START_REF] Blondel | Deterministic nonlinear wave prediction using probe data[END_REF], because of its high numerical efficiency and the accuracy of the results. As already pointed out by [START_REF] Blondel | Deterministic nonlinear wave prediction using probe data[END_REF]; [START_REF] Köllisch | Nonlinear real time prediction of ocean surface waves[END_REF], with conventional HOS models, most of the computational time is spent in determining a valid initial condition for the model, from the collected wave data. This initialization step is the main obstacle to realtime prediction today, because it requires the knowledge of two independent spatial quantities: the free surface elevation and the velocity potential at the free surface. As this is hardly achievable with classical instrumentation, the velocity potential can be approximated at first order [START_REF] Klein | Systematic Experimental Validation of High-Order Spectral Method for Deterministic Wave Prediction[END_REF], introducing errors, or computed using data assimilation [START_REF] Aragh | Variation Assimilating of Synthetic Radar Data into a Pseudo-spectral Wave Model[END_REF][START_REF] Blondel-Couprie | Reconstruction and Prediction of Short-Crested Seas Based on the Application of a 3D-FFT on Synthetic Waves: Part 2 -Prediction[END_REF][START_REF] Yoon | An Explicit Data Assimilation Scheme for a Nonlinear Wave Prediction Model Based on a Pseudo-Spectral Method[END_REF][START_REF] Köllisch | Nonlinear real time prediction of ocean surface waves[END_REF], which greatly increases the computational cost.

Another approach was recently proposed by [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF] and tested numerically. The present paper builds on this previous work. It develops a nonlinear deterministic sea wave prediction method based on fixed-point velocity measurements. Rather than sea surface elevation data sets, which are most widely used [START_REF] Naaijen | Phase Resolved Wave Prediction From Synthetic Radar Images[END_REF][START_REF] Desmars | Experimental and numerical assessment of deterministic nonlinear ocean waves prediction algorithms using non-uniformly sampled wave gauges[END_REF], the method relies on wave-induced velocity measurements to retrieve information on the incoming sea state. An instantaneous, deterministic, horizontal velocity profile is deduced from this measurements. A nonlinear wave model based on the HOS method then propagates the corresponding waves downstream. Unlike other approaches using a HOS model, here we use a formulation which was initially developed for numerical wave tank simulations, and slightly adapted to fit our purpose. As a consequence, while the initialization of the model is usually responsible for prohibitive computational effort, it is much easier here, because the simulation starts from rest and there is no need for data assimilation. In our approach, the critical step is rather to retrieve an instantaneous wave-induced velocity profile from available sensors, to feed into the wave model.

As a first step, numerical studies were conducted by [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF] to assess the performance of this prediction method. Promising results were obtained when using perfect, synthetic velocity data, available in the whole water column. With synthetic data only available in a portion of the water column, the results showed an increase in error but were still highly satisfactory. Provided the horizontal velocity profiles containing information on the sea state were accurately retrieved, the whole prediction method showed high quality results, with a good reproduction of the sea surface elevation in the spatio-temporal domain theoretically accessible to prediction. The accurate restitution of the waves phase information, even for steep sea states, was an important result of these numerical tests, as it confirmed the interest of using a nonlinear wave propagation model for prediction purposes.

Following these results, the present paper reports on the experimental validation of the method on long-crested irregular waves, in a controlled wave tank environment. Velocity data were collected with an ADV (Acoustic Doppler Velocimeter) and two ADCP (Acoustic Doppler Current Profiler) sensors. This is a particularly challenging experimental set-up, because ADCPs require the presence of small particles in the water column to work properly, which are very seldom in wave tanks. However, it allows direct testing of the suitability of these instruments for the prediction method. The quality of the measurements was found to be highly dependent on the experimental conditions (namely, the homogeneous distribution of scatterers in the water column). For trials complying with this requirement, the quality of the horizontal velocity profiles reconstructed from these acoustic measurements was assessed. The prediction obtained after feeding these profiles into the wave model agreed well with the experimental measurements.

The remainder of the present paper is structured as follows: Section 2 describes the prediction method, Section 3 presents the experimental setup and the validation principle of the study conducted. Section 4 provides an analysis of the measurements collected, and Section 5 discusses the results.

Prediction method

The prediction method has already been studied numerically. A detailed description of the method is available in [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF]. Only its essential elements are reminded here for comprehension.

Assumptions and general principle

The study considers long-crested waves propagating over a constant depth ℎ. The effects of wind forcing, wave breaking and ambient currents are not taken into account. The space domain  is defined horizontally between 𝑥 = 𝑥 0 and 𝑥 = 𝑥 0 + 𝐿 𝑥 , and vertically between the flat bottom 𝑧 = -ℎ and the single-valued free surface position 𝑧 = 𝜂(𝑥, 𝑡). We use a Cartesian coordinate system, (𝑂, 𝑥, 𝑧), set as shown in Figure 1: the horizontal axis is located at the mean water level, waves are propagating towards 𝑥 > 0, and the vertical axis is positive-upward. Finally, we use the potential flow theory, i.e. we assume that the fluid is inviscid, incompressible and with an irrotational motion.

In this context, the method discussed here aims to provide a deterministic wave prediction, at a horizon of a few wave periods (around 20 -40 seconds for full scale sea states). Its principle is illustrated in Figure 1. Using a fixed measurement device deployed upstream of the area of interest, the wave-induced flow velocity is collected continuously along the water column, at a position noted 𝑥 0 . The data is used to reconstruct the profile of the horizontal component of the instantaneous fluid velocity: 𝑈 (𝑥 0 , 𝑧, 𝑡). This horizontal velocity profile is then passed on as a boundary condition in a deterministic, nonlinear wave propagation model. Starting from initial rest, the prediction process can then be broken down into two steps: the reconstruction of the sea state and the actual prediction.

• The reconstruction step, of duration 𝑇 , is arbitrarily set in the past and is presented in Figure 1a. For -𝑇 ≤ 𝑡 ≤ 0, the velocity measurements collected at 𝑥 0 in the water column are continuously updated.

The boundary condition provided to the propagation model is obtained from these measurements. It is given as an instantaneous horizontal velocity profile imposed at 𝑥 0 . The model generates and propagates the corresponding waves from 𝑥 0 to the downstream area of interest.

• The prediction step is illustrated in Figure 1b. At 𝑡 > 0, velocity data containing information on the incident waves at 𝑥 0 are no longer updated, and the boundary condition is set to zero in the wave model. Therefore, it propagates only the information collected before 𝑡 = 0 and already present in the simulation domain.

Theoretical prediction zone

Because the information collected about the incident sea state is limited in time and space, the spatio-temporal region over which a deterministic wave prediction can be established using these measurements is also bounded. This region is called the theoretical prediction zone and is noted  𝑥,𝑡 in the present article. The topic has been discussed extensively before, e.g. in [START_REF] Wu | Direct simulation and deterministic prediction of largescale nonlinear ocean wave-field[END_REF]; Naaijen et al. (2014); [START_REF] Qi | Predictable zone for phaseresolved reconstruction and forecast of irregular waves[END_REF]; [START_REF] Fucile | A probabilistic approach for the quantification of prediction error in deterministic phase-resolved wave forecasting[END_REF], so the definition provided below is directly adapted to the present case study. We simply indicate here that an accurate description of the sea state is possible only if the energy content of the wave spectrum is known. This content propagates at the wave group velocity. Therefore, besides the location and duration of wave measurements, the extent of the theoretical prediction zone also depends on the limiting group velocities of the sea state components. Assuming the wave energy is concentrated around the peak frequency of the spectrum, it becomes negligible outside some well-chosen frequency bounds 𝑓 𝑚𝑖𝑛 and 𝑓 𝑚𝑎𝑥 , and the spectrum evolution can be studied within these bounds.

In the present case study, a single, fixed measurement point upstream is located at 𝑥 0 and measures continuously for a period of time 𝑇 . The corresponding theoretical prediction zone  𝑥,𝑡 is defined by :

(𝑥 1 , 𝑡 1 ) ∈  𝑥,𝑡 if -𝑇 + 𝑥 1 -𝑥 0 𝐶 𝑔 𝑚𝑖𝑛 ≤ 𝑡 1 ≤ 𝑥 1 -𝑥 0 𝐶 𝑔 𝑚𝑎𝑥 (1)
where 𝐶 𝑔 𝑚𝑖𝑛 = 𝐶 𝑔 (𝑓 𝑚𝑎𝑥 ) and 𝐶 𝑔 𝑚𝑎𝑥 = 𝐶 𝑔 (𝑓 𝑚𝑖𝑛 ) are the group velocities of respectively the slowest and the fastest components of the considered sea state. At any fixed location 𝑥 1 , the prediction zone is also characterized by its temporal boundaries 𝑡 𝑚𝑖𝑛 (𝑥 1 ) = -𝑇 + (𝑥 1 -𝑥 0 )∕𝐶 𝑔 𝑚𝑖𝑛 and 𝑡 𝑚𝑎𝑥 (𝑥 1 ) = (𝑥 1 -𝑥 0 )∕𝐶 𝑔 𝑚𝑎𝑥 , the latter being the maximum prediction horizon.

The total prediction zone  𝑥,𝑡 can be further divided into a reconstruction sub-region  𝑟𝑒𝑐 for 𝑡 ≤ 0 and a prediction sub-region  𝑝𝑟𝑒𝑑 for 𝑡 > 0, as illustrated in Figure 2b.

Nonlinear wave prediction model

In the present method, the information collected on the incident sea state is used to reconstruct and propagate the waves along (𝑂𝑥), using a horizontal flow imposed at 𝑥 0 . This step is performed with the open-source wave model HOS-NWT 1 . This code is a nonlinear, deterministic Numerical Wave Tank based on the HOS method [START_REF] Touzé | Méthodes spectrales pour la modélisation nonlinéaire d'écoulements à surface libre instationnaires[END_REF][START_REF] Bonnefoy | Modélisation expérimentale et numérique des états de mer complexes[END_REF][START_REF] Ducrozet | Modélisation des processus non-linéaires de génération et de propagation d'états de mer par une approche spectrale[END_REF][START_REF] Bonnefoy | Time domain simulation of nonlinear water waves using spectral methods[END_REF]. As shown by [START_REF] Ducrozet | A modified High-Order Spectral method for wavemaker modeling in a numerical wave tank[END_REF], the use of a Fast Fourier Transforms (FFT) solution coupled to an accelerated scheme makes the code very efficient, and the fast convergence also allows for a high accuracy in the results. HOS-NWT thus appears particularly suitable for the prediction of complex 1 Code available at: https://github.com/LHEEA/HOS-NWT sea states, while maintaining the possibility of reaching realtime prediction, if properly optimized. Moreover, generating waves with a numerical wavemaker is very similar to generating waves by imposing a velocity profile at a fixed location. Only minor changes were thus needed in the model to serve the targeted application. Detailed explanations of the functioning of the HOS-NWT model, as well as validation cases, are available in several works, such as [START_REF] Ducrozet | Implementation And Validation of Nonlinear Wavemaker Models In a HOS Numerical Wave Tank[END_REF][START_REF] Ducrozet | A modified High-Order Spectral method for wavemaker modeling in a numerical wave tank[END_REF]. The minor modifications made to adapt it to our prediction problem are presented in [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF]. In the following, the general features of the code are only reminded for the sake of completeness.

In the original version of HOS-NWT, within the fluid domain  previously defined, the section 𝑥 = 𝑥 0 corresponds to the wavemaker's rest position, and the section 𝑥 = 𝑥 0 + 𝐿 𝑥 represents a perfectly reflective wall. HOS-NWT runs under the potential flow assumptions introduced in section 2.1. The bottom boundary condition is a no-flow condition. The free surface boundary conditions are written in a fully-nonlinear form following [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF]. They are solved at the position of the free surface, using the orderconsistent High-Order Spectral method of [START_REF] West | A new numerical method for surface hydrodynamics[END_REF].

It allows a rapid and accurate resolution of the free surface equations, at an arbitrary order of nonlinearity 𝑀.

The lateral boundary conditions are expressed as Neumann (no-flow) conditions. In our adapted version of HOS-NWT, the horizontal flow 𝑈 (𝑥 0 , 𝑧, 𝑡), which accounts for the generation of waves, is imposed at 𝑥 0 instead of being written at the position of a moving wavemaker. As described in Section 2.1, in the prediction method 𝑈 (𝑥 0 , 𝑧, 𝑡) corresponds to the instantaneous horizontal velocity profile obtained from the measurements at 𝑥 0 . Finally, HOS-NWT also models the presence of an absorbing beach near the reflective wall at 𝑥 = 𝑥 0 + 𝐿 𝑥 [START_REF] Bonnefoy | Time domain simulation of nonlinear water waves using spectral methods[END_REF][START_REF] Ducrozet | A modified High-Order Spectral method for wavemaker modeling in a numerical wave tank[END_REF]. In a prediction context, the beach is modeled as fully absorbing and 𝐿 𝑥 is set long enough to avoid reflection in the domain.

The main advantage of using this adapted HOS-NWT model for prediction purposes is that, while having a fully nonlinear wave propagation model, there is no need for a costly optimization procedure, or data assimilation step: the procedure is simpler than determining proper initial conditions when using an HOS formulation for open domains. To initialize such simulations based on the open HOS formulation, two independent spatial quantities are required: the free surface elevation 𝜂 and the surface velocity potential 𝜙 𝑆 . The second quantity is usually hard to extract from field data. In contrast, in HOS-NWT the waves are continuously generated (from rest) through the boundary condition at 𝑥 0 , and the only quantity needed for this is the instantaneous horizontal velocity profile.

Horizontal velocity reconstruction

Like in many prediction methods, the main challenge is to feed the propagation model with reliable in situ data. Here, this means building a valid boundary condition at 𝑥 = 𝑥 0 . This condition is expressed in the form of instantaneous profiles of the horizontal velocity associated to wave propagation. A benchmark of the sensors available on the market identified ADCPs as the most suitable instruments to measure velocities in the water column. However, as ADCPs measure radial velocities (along the acoustic beam's axis), an instantaneous horizontal velocity profile is not directly available with these instruments [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF]. A method was thus developed to reconstruct this quantity using the current measurement capabilities of ADCPs. It is presented here for unidirectional sea states but could be extended to multi-directional case, using more instruments, to retrieve the profile of the 2D horizontal component at a (single) given point.

Besides the four divergent acoustic beams usually present in a conventional ADCP, some recent models are also equipped with a fifth beam, oriented vertically for a bottommounted instrument. It can measure the instantaneous profile of the vertical component of the velocity with a good resolution, see e.g. Nortek (2022b). The method proposed here uses only the vertical beam of two ADCPs to reconstruct an instantaneous horizontal velocity profile. In the proposed measurement configuration, shown in Figure 3, the two ADCPs are separated by a distance Δ𝑥 in the direction of wave propagation (𝑂𝑥), that is small with respect to the wavelength. They measure two instantaneous velocity profiles, denoted 𝑊 (𝑥 0 ± Δ𝑥 2 , 𝑧, 𝑡). The velocity measurements of both ADCPs are synchronized. Under the assumption of irrotational flow, and assuming vertical velocities are available throughout the whole water column at 𝑥 = 𝑥 0 ± Δ𝑥 2 , an approximation 𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧, 𝑡) of the instantaneous horizontal velocity profile can be written as:

𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧 0 , 𝑡) = Δ𝑧 Δ𝑥 𝑁 𝑧 ∑ 𝑘=1 Δ𝑊 (𝑥 0 , 𝑧 𝑘-1 , 𝑡) + Δ𝑊 (𝑥 0 , 𝑧 𝑘 , 𝑡) 2 + 𝑈 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡) + 𝑂(Δ𝑥 2 ) + 𝑂(Δ𝑧 2 ) (2)
where

• Δ𝑊 (𝑥 0 , 𝑧 0 , 𝑡) = 𝑊 (𝑥 0 + Δ𝑥 2 , 𝑧 0 , 𝑡)-𝑊 (𝑥 0 -Δ𝑥 2 , 𝑧 0 , 𝑡); • 𝑧 𝑟𝑒𝑓 is an arbitrary reference depth for which the horizontal velocity 𝑈 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡) is known;

• 𝑧 𝑘 are the coordinates of the grid points on which the integration is made, dividing the interval [𝑧 𝑟𝑒𝑓 ; 𝑧 0 ] into 𝑁 𝑧 sub-intervals of length Δ𝑧. Here, Δ𝑧 is constant and represents the size of the ADCPs' measurement cells.

For deep water applications, assuming the vertical velocity profiles 𝑊 can be measured until the velocity becomes negligible at 𝑧 = 𝑧 𝑟𝑒𝑓 , the reference velocity 𝑈 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡) can be set to zero without introducing much error. At intermediate depths however, a point velocimeter is useful in addition to the two profilers, to provide the horizontal reference velocity.

The proposed method uses instantaneous data, so the quality of the reconstructed velocity itself does not depend on the time discretization used. However, it assumes an accurate synchronization between the measuring devices.

Experiments

In this experimental validation, irregular, unidirectional waves were generated in a wave tank. The fluid velocity and surface elevation associated to these waves were measured in several locations of the tank. Two sea states characterized by a Pierson-Moskowitz spectrum were investigated. For a first validation of the method, a low characteristic wave steepness was selected: 𝜖 𝑐 = 𝐻 𝑠 ∕𝜆 𝑝 = 1%. Two different peak periods (and thus different 𝜆 𝑝 ∕ℎ ratios) were chosen: 𝑇 𝑝 = 2 s (case A, hereafter) and 𝑇 𝑝 = 3 s (case B). This was the result of a compromise between having large enough wave velocities and enough time before measuring significant reflection from the end of the tank. The corresponding significant wave heights were 𝐻 𝑠 = 6 cm and 𝐻 𝑠 = 12 cm, respectively. For each sea state, ten phase sets were tested and lasted 350 s each.

The experiments were carried out to assess the feasibility of: i) finely measuring instantaneous vertical velocity profiles due to wave propagation, using ADCPs; ii) reconstructing instantaneous, wave-induced horizontal velocity profiles from such ADCP data; and iii) obtaining a good quality wave prediction, using these profiles as inputs in the investigated method.

To do so, surface elevation was measured using 17 resistive wave gauges. An ADV collected single-point, 3D velocity measurements at a single position (𝑥 0 , 𝑧 𝑟𝑒𝑓 ) in the tank, and two bottom-mounted, upward-looking ADCPs deployed on either side of 𝑥 0 (in the propagation direction) measured two distinct vertical velocity profiles. Figure 4 shows a schematic view of the general set-up.

Relatively few studies have considered the deployment of ADCPs in a tank filled with fresh water, even less for wave measurements. These studies mostly investigated averaged flow quantities, like [START_REF] Muste | Near-Transducer Errors in ADCP Measurements: Experimental Findings[END_REF], or the grain size distribution of suspended sediments, as in [START_REF] Guerrero | Laboratory validation of acoustic Doppler current profiler (ADCP) techniques for suspended sediment investigations[END_REF]. They highlighted that, when measuring in a fresh water tank, a major difficulty is the non-homogeneous distribution of scatterers in the water column. Depending on the application, it is sometimes possible to use adapted facilities, such as large-scale towing tanks or tanks equipped with a recirculation and mixing system. The wave tank used in the present study does not benefit from such adaptations, and the wave measurements conducted therefore represent a very challenging setup for the use of ADCPs.

Experimental set-up

The experiments were conducted in the towing tank of Centrale Nantes, which is 140 m long, 5 m wide and 3 m deep. Waves are generated at 𝑥 = 0 by a hinged flap rotating around an axis located at 50 cm from the bottom of the tank, and an absorbing beach limits their reflection at the other end of the tank. This beach is composed of a 5 mlong, 20% concrete slope followed by a parabolic, 3.5 m-long perforated part. Two water lines are also laid down near the beach to further reduce wave reflection. The water depth at rest is ℎ = 2.90 m.

Resistance-type wave probes were installed downstream to the wave maker and recorded free surface elevation at 64 Hz, every 6 m in the propagation direction: from 𝑥 𝑝1 = 24 m to 𝑥 𝑝17 = 120 m. They were located far enough from the side wall to avoid spurious reflection and were calibrated before the trials.

Velocity measurements were collected by a single-point velocimeter (ADV), fixed at (𝑥 0 , 𝑧 𝑟𝑒𝑓 ) = (20 m, -0.54 m), and by two bottom-mounted acoustic profilers (ADCPs). These profilers were deployed at 𝑥 𝐴𝐷𝐶𝑃 1 = 𝑥 0 + Δ𝑥∕2 and 𝑥 𝐴𝐷𝐶𝑃 2 = 𝑥 0 -Δ𝑥∕2. The position 𝑥 0 = 20 m was set as close as possible from the wavemaker, but away from the evanescent modes, see e.g. [START_REF] Molin | Hydrodynamique des structures offshore[END_REF].

The ADV measuring the "reference" horizontal velocity time series along 𝑂𝑥 (with the 2 other components of the velocity) was a Nortek Vectrino. It is equipped with a fourbeam head emitting convergent acoustic beams, and it is well-adapted to laboratory use. The instrument's body was fixed to a pillar immersed in the tank, while the rod bearing the measuring head with the transmitters allowed measuring far enough from the structure for the flow not to be influenced by its presence. Measurements were collected at (𝑥 0 , 𝑧 𝑟𝑒𝑓 ) = (20 m, -0.54 m) with a sampling frequency of 16 Hz.

The ADCPs measured instantaneous vertical velocity profiles and were deployed at the bottom of the tank. Their centers were separated by Δ𝑥 = 5 cm in the wave propagation direction. This is the result of a compromise between minimizing the error in Eq. ( 2) and the capabilities of the experimental setup, which must be able to accurately measure the vertical velocity difference Δ𝑊 = 𝑊 (𝑥 0 +Δ𝑥∕2)-𝑊 (𝑥 0 -Δ𝑥∕2). In the transverse direction, the ADCPs were separated by Δ𝑦 = 20 cm, their diameter being too large (15 cm) to obtain the chosen Δ𝑥 spacing if keeping them aligned in the propagation direction.

The ADCPs used were Nortek Signature 1000 kHz. They were equipped with a fifth central beam, in addition to the classical 4 diverging beams. These instruments are usually deployed at sea and were not designed for laboratory trials. The measurement method remains valid, and with an adapted set-up to take this point into account, this allows to directly test the instruments' capabilities. Nevertheless, this aspect is emphasized because it is likely the cause of some difficulties that will be discussed further on.

In these trials, the 4 lateral beams were deactivated to avoid interference and only the central beam was used to measure the vertical velocity 𝑊 (𝑥 0 ± Δ𝑥∕2, 𝑧, 𝑡). This central beam offers a High Resolution mode using a pulsecoherent method. It provides a much higher resolution and a better accuracy, at the price of a reduced achievable velocity range and / or profiling range [START_REF] Rusello | A Practical Primer for Pulse Coherent Instruments[END_REF]. This is not an issue at tank scale. Taking into account the blanking distance and the potential signal degradation near the free surface, the ADCPs provided velocity measurements between 𝑧 𝑚𝑖𝑛 = -2.46 m and 𝑧 𝑚𝑎𝑥 = -0.08 m, with a cell size Δ𝑧 = 2 cm. Their sampling frequency was set to 8 Hz2 .

For all instruments, the start of the recording was triggered by the start of the wavemaker, ensuring the synchronization of the collected data. These were measured for 350 s after the start of the wavemaker: this is long enough to ensure that the sea state is completely established and that the waves have propagated to the area of interest, but it is still shorter than the repetition period of the wavemaker motion, which is set to 512 s. After each experiment, the wavemaker was stopped during 15 minutes to ensure a calm initial state for the following run.

Finally, ADCPs and ADVs are acoustic instruments and require backscatter from suspended particles to work properly. When using such instruments in a clear water tank, additional seeding particles are then required to ensure a sufficient return signal. Here, 10 µm glass micro-spheres with a slightly negative buoyancy were rained down from the top of the water column around 𝑥 0 . These particles were particularly suitable for ADV measurements, but proved less efficient with ADCPs, as will be seen in Section 5.

Validation principle

The measurements made in the wave tank are used in two ways. First, the velocity data collected, 𝑊 𝑒𝑥𝑝𝑒 (𝑥 0 ± Δ𝑥∕2, 𝑧, 𝑡) and 𝑈 𝑒𝑥𝑝𝑒 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡), are used to reconstruct a horizontal velocity profile denoted 𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧, 𝑡), following Equation (2). This experimental reconstruction is validated using a numerical twin, and is fed as input in the prediction method. Second, the wave probes' measurements 𝜂 𝑒𝑥𝑝𝑒 (𝑥 𝑝 , 𝑡) allow to evaluate the quality of the predicted free surface elevation 𝜂 𝑝𝑟𝑒𝑑 (𝑥, 𝑡), obtained with the investigated method.

Regarding the velocity validation, a digital twin was created for each trial to provide synthetic, reference values for the quantities of interest in the whole domain. This digital twin was generated thanks to a feature in the original version of the HOS-NWT code, which allows to specify the exact same wavemaker motion in a numerical wavetank as in the physical experiment [START_REF] Bonnefoy | Time domain simulation of nonlinear water waves using spectral methods[END_REF][START_REF] Ducrozet | A modified High-Order Spectral method for wavemaker modeling in a numerical wave tank[END_REF]. Using this HOS-NWT-twin, each experiment was accurately numerically modeled. The synthetic, reference horizontal velocity 𝑈 𝑛𝑢𝑚 (𝑥 0 , 𝑧, 𝑡) was used to assess the quality of the instantaneous profile 𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧, 𝑡), reconstructed from experimental velocity data.

This instantaneous horizontal velocity profile obtained from experimental data was then used to generate and propagate the waves in the prediction model described in Section 2.3, denoted HOS-NWT-pred. This is the version of the HOS-NWT code adapted to prediction purposes. The results obtained with the prediction model fed by experimental velocity profiles, denoted 𝜂 𝑝𝑟𝑒𝑑 (𝑥, 𝑡), were compared to the surface elevation data 𝜂 𝑒𝑥𝑝𝑒 (𝑥 𝑝 , 𝑡) measured by the wave gauges, to validate the whole prediction method.

The internal time step for both types of HOS-NWT simulations was around 0.02 s. The velocity profile was linearly interpolated from the reconstructed boundary condition in the prediction simulations.

Figure 5 presents a summary of the quantities measured and their use in the validation procedure.

In the prediction simulations, the velocity profiles were given as an imposed horizontal flow for the boundary condition at 𝑥 0 . The corresponding sea state was reconstructed for 𝑇 = 30 𝑇 𝑝 and an additional 5 𝑇 𝑝 ramp accounting for the start of the wave maker. In the common reference frame aligned on the physical tank, the wave generation was forced at 𝑥 0 = 20 m in the prediction model, meaning the prediction simulations were shifted in space with respect to the physical experiment and the reference simulations (which started at 𝑥 = 0). The calculation of the theoretical prediction area was adapted accordingly and the length of the simulation domain in prediction simulations was set to 𝐿 𝑥 = 120 m.

Analysis

Reflection in the wave tank

The presence of reflection in the wave tank is evidenced by the increasing difference, over time, between the free surface measurements and the reference simulation. When restricted to periods of time before the reflected component The velocity reconstruction, using digital twin 1

The prediction method, using experimental data 2 Assessing:

Instructions

Prediction

Figure 5: Overview of the experimental validation process. Each case is characterized by its frequency spectrum and associated set of phases, and the instructions are used: i) by the physical wavemaker to generate the sea state in the wave tank and ii) as input to a digital twin, numerically replicating the trial for validation purposes. The reconstructed horizontal velocity is fed as input to the prediction model.

of frequency 𝑓 𝑚𝑎𝑥 reaches each wave gauge, the analysis shows that the measurements agree very well with the results of the digital twin: the mean absolute difference in free surface elevation (averaged over all wave gauges) is 1.1% of 𝐻 𝑠 for case A.10, and 1.2% of 𝐻 𝑠 for case B.1. However, when considering data over a duration which includes reflection, the mean absolute difference increases respectively to 4.8% and 6.5% of 𝐻 𝑠 . Because the probe spacing is regular and very close to a multiple of 𝜆 𝑝 ∕2 for both considered sea states, reflection in the experimental data could not be corrected ex-post using the method proposed by [START_REF] Mansard | The Measurement of Incident and Reflected Spectra Using a Least Squares Method[END_REF].

Dedicated tests with a single regular wave train evaluated the reflection coefficient for regular waves: with a period 𝑇 = 2 s (resp. 3 s) and an amplitude 𝐴 = 3 cm (resp. 6 cm), a reflection of 10% (resp. 12%) in amplitude was measured.

A complementary approach, adapted from Desmars et al. (2020), measured the residual perturbations in the tank after the wave maker had stopped and the last wave components had propagated past all probes, at a time denoted 𝑡 𝑛 . Assuming that the situation is representative of the whole experiment, the recordings of the free surface elevation from 𝑁 probes were used to calculate a "noise level" 𝐻 𝑛 , built on the same model as the significant wave height 𝐻 𝑠 :

𝐻 𝑛 = 1 𝑁 𝑁 ∑ 𝑖=1 4𝜎 𝜂 (𝑥 𝑖 , 𝑡 ≥ 𝑡 𝑛 ) (3)
The noise-to-signal ratio 𝑁𝑆𝑅 = 𝐻 𝑛 ∕𝐻 𝑠 derived from this analysis was estimated at 10% for case A, and at 9% for case B. In [START_REF] Desmars | Experimental and numerical assessment of deterministic nonlinear ocean waves prediction algorithms using non-uniformly sampled wave gauges[END_REF], for the same wave steepness the measured NSR was around 20%. This may be explained by the fact that their experiment was conducted in a wider wave tank, more likely to generate transverse disturbances because of the wavemaker flaps.

The frequency distribution of the noise was also estimated by computing the power spectral density 𝑆 𝑛 of the noisy part of the signal, obtained by averaging the results over the 𝑁 wave probes. The normalized noise spectrum 𝑆 * 𝑛 = 𝑆 𝑛 𝑓 𝑝 ∕𝑚 0 , with 𝑚 0 = 𝐻 𝑛 ∕16, is presented in Figure 6 for both sea states. The normalized target wave spectrum is also provided for illustrative purposes. The reflection maximum frequency is around 𝑓 0 = 0.44 Hz in both sea states, i.e. 0.9𝑓 𝑝 for case A and 1.3𝑓 𝑝 for case B. The reflected energy is more concentrated around the peak for case A than for case B. In both cases, noise at low frequencies is present but at a much lower level than around the peak. It can be mitigated by processing the data with a high-pass filter, provided the cut-off frequency does not mask information related to energetic components of the wave spectrum.

As the effects of reflection are particularly important around 𝑓 0 , whenever possible the analysis was restrained to the instants preceding the arrival of the reflected wave of frequency 𝑓 0 at the point of interest. This is the case when evaluating the quality of the reconstructed horizontal velocity profile: the position of the instruments around 𝑥 0 , i.e. close to the wavemaker, allows a quick establishment of a complete sea state and the measurements duration is thus sufficient to analyze the results. Unfortunately, this is not possible for the prediction results. This is because the time needed to establish a complete sea state at the most downstream probes (which are used to evaluate the prediction on 𝜂) is greater than the travel time of the reflected wave of frequency 𝑓 0 (which is the time taken by the frequency component 𝑓 0 to reach the downstream probes after reflection on the beach).

Data quality control

Overview

The wave gauges were calibrated before the trials and their use in this experimental setup is classical: therefore, applying a perfect low-pass filter at 𝑓 𝑐𝑢𝑡 = 8 Hz is sufficient after data collection.

For ADV as well as ADCP measurements, in addition to raw velocity data the instruments also provide auxiliary quantities that help evaluate the data quality. One of them is the correlation, which is "a statistical measure of how similar the received signal is to itself at a delayed time" (Nortek, 2022b). The exact definition of correlation varies depending on the instruments and on manufacturers, but the principle remains the same. The higher the correlation, the most reliable the data, because it means the instrument is more likely to have measured correctly the pulses it emitted, and determined a valid velocity. Drops in correlation allow to identify events when the signal can be too weak or too blurred to allow collecting good data.

Here, the seeding system designed to provide a high concentration of scatterers in the wave tank worked very well for the ADV measurements, as expected since this instrument is routinely used in laboratory experiments. The correlation values associated to the ADV measurements remain higher than 80% for all beams from all trials, when the advised minimum threshold is 70% in usual applications (Nortek, 2022a). It is even higher than 95% for more than 97% of measurements from beam 1 3 , indicating excellent data quality. For this ADV data, a (perfect) low-pass filter at 𝑓 𝑐𝑢𝑡 = 8 Hz is also sufficient.

In contrast, the quality of the vertical velocity profiles measured by the ADCPs in the wave tank environment is overall disappointing. Figure 7 shows an example of raw ADCP data: the first subplot presents the vertical velocities measured by the instrument and the bottom subplot shows the values of associated correlation. Generally, correlation 3 According to the Vectrino's orientation, this beam directly provides the horizontal velocity.

is used in quality control to identify the reliable data range. It also helps spotting outliers, discarding any velocity data point whose associated correlation falls below a user-defined threshold. As can be seen in Figure 7, the data quality is unequal, very poor in some areas, and calls for a closer look.

Processing of ADCP data

Numerous outliers are visible in the data, and the signal quality is not homogeneous. It varies with both the trial and the depth, with a particularly poor data quality at the bottom of the water column (very low correlation). This means this is not a problem of measurement range. Besides, despite legitimate concerns due to the proximity between the ADCPs, there were no recognizable signs of interference between the beams. These would translate into distinct and consistent areas of poor correlation; what is observed here looks more like patterns induced by flocks of slowly moving scatterers, as shown by the stratified patterns.

Regardless of the application, ADCP data need to undergo a proper quality control procedure before they can be analyzed. This is especially true here, as we need the data to be accurate enough to reconstruct an instantaneous horizontal velocity profile, which should describe the deterministic influence of waves in the water column. Here, the signal is too noisy for a classical spectral filtering to be useful.

The quality check conducted started by removing all data points associated with a correlation lower than 80%. Then, because phase-resolved wave prediction is possible only if a sufficient amount of information is available from both ADCPs at the same time, only trials where both ADCPs passed the correlation test more than 90% of the time were kept. This left only one suitable case for 𝑇 𝑝 = 2 s: A.10, and two for 𝑇 𝑝 = 3 s: B.1 and B.3. For the sake of conciseness, only cases A.10 and B.1 are presented in the following.

Another outlier detection round was conducted cell by cell, using first the criterion |𝑊 𝑜𝑢𝑡𝑙𝑖𝑒𝑟 (𝑧, 𝑡) -⟨𝑊 (𝑧)⟩| > 6 𝜎 𝑊 (𝑧), then the Phase-Space Thresholding Method (PSTM). It is an iterative detection method that was first proposed by [START_REF] Goring | Despiking Acoustic Doppler Velocimeter Data[END_REF] for the despiking of ADV data time series. It uses 3D Poincaré maps and the first two time derivatives of the signal. The incorrect data points spotted were replaced by interpolation. Finally, the resulting 𝑊 arrays were smoothed by vertically fitting a 𝑠𝑖𝑛ℎ-type function on each instantaneous profile. More detail on the whole procedure can be found in Appendix A. What can be noted here, however, is that the processing steps conducted in the procedure described are not yet compatible with realtime, as for example the PSTM is applied a posteriori.

This specific treatment is more intensive that what is usually done for ADCP data, but is necessary here due to the challenge of collecting good quality data in a lab environment with fresh clear water. Data quality control in real conditions is expected to be more simple, and should be possible with conventional techniques for raw ADCP data (see e.g. [START_REF] Furgerot | One year of measurements in Alderney Race: preliminary results from database analysis[END_REF]), which rely on ancillary quantities such as correlation. This type of data is provided from the instrument's internal processing, as a direct output of the ADCP measurements. It is thus available at the same time as the velocity data itself and requires no additional processing time.

Horizontal velocity reconstruction

After data processing, the instantaneous horizontal velocity profile 𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧, 𝑡) is computed from the singlepoint horizontal velocity measurements 𝑈 𝑒𝑥𝑝𝑒 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡) provided by the ADV, and the two vertical velocity profiles 𝑊 𝑒𝑥𝑝𝑒 (𝑥 0 ± Δ𝑥∕2, 𝑧, 𝑡) measured by the ADCPs. Here, preliminary tests have shown that using 𝑊 data below 𝑧 𝑟𝑒𝑓 did not improve the quality of the reconstruction. This is mostly due to the inconsistent quality of ADCP measurements, especially further from the surface. Besides, the integration error on 𝑧 accumulates when moving away from 𝑧 𝑟𝑒𝑓 . The horizontal velocity profile is then reconstructed using Equation (2) only between 𝑧 𝑟𝑒𝑓 and 𝑧 𝑚𝑎𝑥 . Discarding data below 𝑧 𝑟𝑒𝑓 means that only |𝑧 𝑟𝑒𝑓 |∕ℎ ≈ 18% of the profile is sampled experimentally4 . However, this is where the most important information on waves is found. In the rest of the profile, a vertical fitting is run for each time step, where the fitting function is a 3-component approximation of the expression of the horizontal velocity in the linear wave theory. Finally, a perfect high-pass filter with a cutoff frequency 𝑓 𝑐 = 0.1 Hz is applied to the reconstructed horizontal velocity, removing any low frequencies present in the signal that would not be useful for describing the sea state. Note that applying a perfect (Fourier) filter is not compatible with real-time data processing, as it is noncausal. This would need to be improved in the future.

Error indicators definitions

As stated in Section 4.2, the quality of velocity data is variable because the scatterers concentration was not homogeneous in the measurement area, despite the seeding system being designed to minimize this problem. This led to noisy measurements, and thus to a high proportion of missing data in some areas. Analysis of the experimental data showed that this heterogeneity is visible within a single trial: the upper part of the water column is better covered than the bottom. But experimental conditions also changed from one tank test to another. Consequently, differences in data quality are found between trials. In the following, the analysis of the results is then restricted to the trials for which the experimental conditions allowed the best quality data to be collected: one for each sea state tested. This reduces the number of factors influencing the results and makes the analysis more simple.

On velocity reconstruction

The reconstructed horizontal velocity profile, 𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧, 𝑡), is compared with the data provided by the reference simulations, 𝑈 𝑛𝑢𝑚 (𝑥 0 , 𝑧, 𝑡). Parasitic reflections at the end of the physical wave tank are difficult to reproduce accurately in these simulations (see Section 4.1). To minimize their influence, the results of the velocity reconstruction are thus analyzed only for the period 𝑇 limited by:

• the time when a complete sea state is established at 𝑥 0 : 𝑡 𝑚𝑖𝑛 = 𝑥 0 ∕𝐶 𝑔 (𝑓 𝑚𝑎𝑥 );

• the time when the waves of frequency 𝑓 𝑝 arrive at 𝑥 0 after being reflected by the beach at 𝑥 = 𝑥 0 + 𝐿 𝑥 , which are likely to be the main contributor to the parasite waves:

𝑡 𝑚𝑎𝑥 = (2𝐿 𝑥 -𝑥 0 )∕𝐶 𝑔 (𝑓 𝑝 ).
With 𝑓 𝑚𝑎𝑥 selected to neglect the components responsible for less than 5% of the maximal spectrum density, this corresponds to a total reconstruction period 𝑇 = 126 s ≈ 63 𝑇 𝑝 for cases A (𝑇 𝑝 = 2 s), and 𝑇 = 70 s ≈ 23 𝑇 𝑝 minimum for cases B (𝑇 𝑝 = 3 s). The error indicator selected provides a normalized error profile averaged over this time range, during which reflection should be minimal:

𝜖 𝑁𝑅𝑀𝑆 (𝑧) = [ ∫ 𝑇 0 [ 𝑈 𝑟𝑒𝑐 (𝑥 0 , 𝑧, 𝑡) -𝑈 𝑛𝑢𝑚 (𝑥 0 , 𝑧, 𝑡) ] 2 𝑑𝑡 ∫ 𝑇 0 𝑈 𝑛𝑢𝑚 (𝑥 0 , 0, 𝑡) 2 𝑑𝑡 ]1∕2 (4)
where 𝑈 𝑟𝑒𝑐 is the reconstructed velocity, 𝑈 𝑛𝑢𝑚 is the velocity provided by the digital twin, and 𝑇 is the time range considered for the reconstruction in each trial.

On wave prediction

This section introduces several quality metrics for the deterministic prediction of free surface elevation. Some of them are classical error indicators, others were provided to allow the comparison of our results to previous studies.

Following [START_REF] Desmars | Experimental and numerical assessment of deterministic nonlinear ocean waves prediction algorithms using non-uniformly sampled wave gauges[END_REF], the accuracy of the predicted wave field is assessed with the misfit indicator (𝑥, 𝑡) :

(𝑥, 𝑡) = 1 𝑁 𝑠 𝑁 𝑠 ∑ 𝑖=1 |𝜂 𝑝𝑟𝑒𝑑,𝑖 (𝑥, 𝑡) -𝜂 𝑒𝑥𝑝𝑒,𝑖 (𝑥, 𝑡)| 𝐻 𝑠 (5)
where 𝜂 𝑝𝑟𝑒𝑑 is the predicted surface elevation, 𝜂 𝑒𝑥𝑝𝑒 is the surface elevation measured by the wave probes and 𝑁 𝑠 is the number of samples used to get an average misfit. Ideally the 𝑁 𝑠 samples should be completely independent, i.e. recorded from different wave fields with random phases sets. Because of experimental costs and of the varying experimental conditions, this was not possible in this study. Instead, here the different phase sets are replaced by 𝑁 𝑠 samples from a single trial, taken as partially overlapping sub-segments of the time series. Each sub-segment is shifted from the previous one by a duration Δ𝑡 < 𝑇 . It has been shown (e.g. in Naaijen et al. (2014) and [START_REF] Desmars | Experimental and numerical assessment of deterministic nonlinear ocean waves prediction algorithms using non-uniformly sampled wave gauges[END_REF]) that this method allows statistical averaging, provided the total measurement period is long enough and reflection does not affect the results too much.

The information used to evaluate the misfit (𝑥, 𝑡) then covers a total duration of 𝑇 𝑡𝑜𝑡 = 𝑇 + (𝑁 𝑠 -1)Δ𝑡. A convergence study (not shown here) determined that in our case, with Δ𝑡 = 1 s the prediction error (𝑥, 𝑡) is fully converged for 𝑇 𝑡𝑜𝑡 ≈ 60𝑇 𝑝 , but is already stable from 𝑇 𝑡𝑜𝑡 ≥ 50𝑇 𝑝 , which is then the criterion defined here. This is important because in this set-up, the longer we record the signal, the more important reflection problems become.

From the error field (𝑥, 𝑡), a prediction error at any given location 𝑥 is obtained by averaging (𝑥, 𝑡) over the prediction zone :

  (𝑥) = 1 𝑡 𝑚𝑎𝑥 (𝑥) -𝑡 𝑚𝑖𝑛 (𝑥) ∫ 𝑡 𝑚𝑎𝑥 𝑡 𝑚𝑖𝑛 (𝑥, 𝑡)𝑑𝑡 (6)
where 𝑡 𝑚𝑖𝑛 (𝑥) and 𝑡 𝑚𝑎𝑥 (𝑥) are respectively the beginning and the end of the theoretical prediction zone at the considered location.

Still following [START_REF] Desmars | Experimental and numerical assessment of deterministic nonlinear ocean waves prediction algorithms using non-uniformly sampled wave gauges[END_REF], a complementary metrics is also provided in the form of a cross-correlation indicator. Cross-correlation is well adapted to deterministic comparison because it measures the similarity between two time series : here, the free surface elevation recorded by a given probe and the corresponding prediction at this location. At any point 𝑥 𝑝 , the time-averaged cross-correlation is defined by:

𝐶(𝜏) = 1 𝑡 𝑚𝑎𝑥 -𝑡 𝑚𝑖𝑛 ∫ 𝑡 𝑚𝑎𝑥 𝑡 𝑚𝑖𝑛 𝜂 * 𝑝𝑟𝑒𝑑 (𝑥 𝑝 , 𝑡)𝜂 * 𝑒𝑥𝑝𝑒 (𝑥 𝑝 , 𝑡+𝜏)𝑑𝑡 (7)
where 𝜂 * 𝑝𝑟𝑒𝑑 = 𝜂 𝑝𝑟𝑒𝑑 ∕𝜎 𝜂 𝑝𝑟𝑒𝑑 is the free surface elevation predicted by the investigated method, normalized by its standard deviation over the (temporal) prediction zone, and 𝜂 * 𝑒𝑥𝑝𝑒 = 𝜂 𝑒𝑥𝑝𝑒 ∕𝜎 𝜂 𝑒𝑥𝑝𝑒 similarly refers to the normalized free surface elevation measured by the wave probes. The calculation gives the correlation factor 𝐶 as a function of a time delay 𝜏. Its maximum 𝐶 𝑚𝑎𝑥 , and the time lag 𝑇 𝑠 for which it is reached, allow to estimate respectively the similarity in shape and amplitude between the signals, and the phase shift of one with respect to the other. The crosscorrelation of a signal with itself will thus give a maximum amplitude 𝐶 𝑚𝑎𝑥 = 1 for a zero time lag. Here, for each case considered, 𝐶 𝑚𝑎𝑥 and 𝑇 𝑠 are calculated for all the 𝑁 𝑠 predictions obtained from the data sub-samples, and averaged over these 𝑁 𝑠 simulations for each probe location.

Finally, to allow further comparisons with previous studies, another classical indicator is provided, relying on correlation as well. As in (5), it is a field indicator, expressed as:

𝑐(𝑥, 𝑡) = 1 𝑁 𝑠 𝑁 𝑠 ∑ 𝑖=1 𝜂 * 𝑝𝑟𝑒𝑑,𝑖 (𝑥, 𝑡) 𝜂 * 𝑒𝑥𝑝𝑒,𝑖 (𝑥, 𝑡) (8)
where, unlike in (7), 𝜎 𝜂 the standard deviation of 𝜂 is computed over the whole measurement time and not only within the [𝑡 𝑚𝑖𝑛 ; 𝑡 𝑚𝑎𝑥 ] time range.

Results

This section presents the findings of the experimental validation: first for the reconstruction of the horizontal velocity profile, then for the prediction of free surface elevation. As a reminder, because of diverging data quality between trials, the analysis of the results is restricted to one trial per sea state.

Horizontal velocity profile

After the data quality control procedure, trials A.10 and B.1 still show a high proportion of valid ADCP data near the free surface for both instruments. They are therefore interesting examples, especially as they were carried out one right after the other and the experimental conditions are likely to be very similar between these two trials.

Figure 8 presents the error profiles obtained for these cases, respectively for case A.10 (𝑇 𝑝 = 2 s, (a)) and case B.1 (𝑇 𝑝 = 3 s, (b)). This error is obtained by comparing indirect experimental results (i.e. the velocity reconstructed from experimental data) with the reference 𝑈 𝑛𝑢𝑚 from HOS-NWT-twin. The coloured area indicates the depth range over which the ADCP data were used. For each case, three profiles are shown:

• The dashed orange profile (

) is obtained by using only measurements in the reconstruction: vertical velocities 𝑊 𝑒𝑥𝑝𝑒 (𝑥 0 ± Δ𝑥∕2, 𝑧, 𝑡) provided by the two ADCPs for 𝑧 ∈ [ 𝑧 𝑟𝑒𝑓 ; 𝑧 𝑚𝑎𝑥 ] ; reference horizontal velocity 𝑈 𝑒𝑥𝑝𝑒 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡) given by the ADV.

• The solid gray profile (

) is obtained by using only synthetic data from the digital twin, with the same "measurement" configuration as for the experimental setup (i.e. 𝑊 𝑛𝑢𝑚 data restricted to 𝑧 ∈ [ 𝑧 𝑟𝑒𝑓 ; 𝑧 𝑚𝑎𝑥 ] ). It represents the minimum error theoretically achievable with the reconstruction method for a given measurement configuration, assuming perfect data. It also highlights the consequences of data not being available outside [ 𝑧 𝑟𝑒𝑓 ; 𝑧 𝑚𝑎𝑥 ] , without the influence of measurement errors.

• The dashed blue profile (

) is obtained by replacing the data from the ADV by the data provided by the digital twin: 𝑈 𝑛𝑢𝑚 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡). This eliminates the measurement error of the ADV from the evaluation of the reconstructed profile, and allows to focus solely on the performance of the ADCPs.

The results obtained with purely experimental data show that for both cases, the error is minimal around 𝑧 = 𝑧 𝑟𝑒𝑓 and increases both upwards and downwards from there. Towards the surface, the increase is sharp, while it is smoother towards the bottom.

Around 𝑧 = 𝑧 𝑟𝑒𝑓 , the horizontal velocity is directly provided by the ADV. The fully experimental error (in orange) does not reach zero however, because it compares ADV measurements with the horizontal velocity provided by the digital twin: the discrepancy between measurements and synthetic data leads to an error of 6.2% at 𝑧 𝑟𝑒𝑓 for case A.10, and 11% for case B.1. Because the NRMS error is quadratic, a comparison of time series at this depth (not shown here) would show that these error levels actually reflect a very good agreement between the two quantities.

The synthetic (gray) and semi-experimental (blue) error profiles never reach zero either, even at 𝑧 𝑟𝑒𝑓 . This is due to the vertical fitting step (see Section 4.2.2), which makes the velocity at 𝑧 𝑟𝑒𝑓 slightly different from 𝑈 𝑛𝑢𝑚 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡) in these profiles. It is also responsible for slightly shifting all error minima away from 𝑧 𝑟𝑒𝑓 .

In the lower part of the water column, the experimental and the synthetic error profiles are similar: in this area, the error is mainly driven by the need for extrapolation caused by missing data. As already shown in [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF], the quality of the reconstruction at these depths is however not the main problem: around |𝑧|∕𝜆 𝑝 ≈ 0.3, the velocity amplitudes involved are lower and contain less useful information for the generation of the sea state than in the upper part of the water column.

Between 𝑧 𝑟𝑒𝑓 and 𝑧 𝑚𝑎𝑥 , both the fully experimental (orange) and the semi-experimental (blue) error profiles show a sharp increase: up to 53% for case A.10 and to 35% for case B.1, at 𝑧 = 0, for the experimental error. This confirms that the precision of ADCP measurements is a determining factor in the quality of the reconstructed velocity, as expected. By contrast, the purely synthetic error (in gray) barely varies in this range where ADCP velocity measurements are available. The sudden increase of the synthetic error close to the free surface is due to the lack of data in the last cm below 𝑧 = 05 .

Finally, close to the free surface, the experimental error is higher for 𝑇 𝑝 = 2 s than for 𝑇 𝑝 = 3 s. The velocities involved are lower for 𝑇 𝑝 = 2 s, so measurement errors on 𝑊 𝐴𝐷𝐶𝑃 1,2 are proportionally greater, especially for small orbital amplitudes. Besides, with no measurements available right below the free surface, we are blind to small wavelengths, which results in a poor resolution of very short waves. This has a greater impact on sea states with small peak periods, because such waves are proportionally more important to accurately describe these sea states. This phenomenon is also visible in the synthetic error profiles.

The error difference is illustrated in Figure 9, which presents time series of reconstructed horizontal velocities at 𝑧 = 0, for 𝑇 𝑝 = 2 s and for 𝑇 𝑝 = 3 s. The difficulties in resolving short waves are clearly visible (e.g. around 𝑡∕𝑇 𝑝 ≈ 37 𝑇 𝑝 for case A.10), and are more important in case A.10 than in case B.1. Even with an error around 55% however, the main patterns of the velocity signal are correctly reproduced for the larger waves. This suggest the relatively high error value accounts for the poor resolution of small waves.

In a real-scale deployment, ADCP measurement errors of this magnitude are anyway unlikely. Indeed, using an ADCP in a reduced-scale, freshwater tank represents a rather difficult case study, compared to normal operating conditions. However, in our wave tank context it remains to be seen what quality of prediction can be reached with the velocity profiles reconstructed under such experimental conditions. This topic is explored in the next subsection.

Wave prediction

To give a first overview of the results, time series of surface elevation predictions are compared to wave probe measurements for both sea states studied, in Figure 10 (case A.10) and Figure 11 (case B.1). In each case, the data subsegment used as an illustration was selected to be consistent with the mean characteristics of the obtained prediction. These mean characteristics are presented in the following figures: Figure 12 shows the misfit indicator (𝑥, 𝑡), Figure 13 shows the correlation map 𝑐(𝑥, 𝑡) as an alternative metrics, Figure 14 shows the mean prediction error at each probe   (𝑥 𝑝 ),and Figure 15 shows the overall correlation magnitude 𝐶 𝑚𝑎𝑥 and associated time shift 𝑇 𝑠 .

Overall, the predicted free surface elevation agrees well with the wave probes' measurements in the prediction zone.

Discrepancies are visible for small amplitude waves. Figure 10 and 11a show examples of very good agreement between the predicted and the measured waves, although case A.10 shows a tendency to underestimate wave amplitudes by around 4% of 𝐻 𝑠 . This behaviour is consistent with the underestimation of the reconstructed velocity amplitudes already visible in Figure 9a. This bias was not detected for case B.1. The missing velocity data between 𝑧 𝑚𝑎𝑥 and 𝑧 = 0 are likely the cause of this phenomenon, as it means short waves are not properly accounted for, especially with 𝑇 𝑝 = 2 s.

In the representative example time series displayed in Figure 10 and Figure 11, the quality of the prediction decreases outside the prediction zone, as information on frequency components lower than 𝑓 𝑚𝑖𝑛 = 0.67𝑓 𝑝 are not available any more. This is consistent with the spatio-temporal evolution of the average misfit presented in Figure 12, which confirms that the prediction zone is a relevant indicator of where a reliable prediction can be generated. The figure suggests that the frequency boundaries selected for this case study might even be too conservative, especially for case A.10. Even outside the prediction zone, the observed deterioration is progressive: except when small and very irregular waves are present like in Figure 11b, the method correctly reproduces the arrival instants of the main waves, even some time after the maximum theoretical prediction horizon.

Looking at averaged parameters, Figure 14 shows that the mean prediction error 𝜀  (𝑥) provided by the new prediction algorithm lies between 5 and 9.5% of 𝐻 𝑠 , depending on the sea state and the position of the probes considered. Case A.10 shows a slight but rather constant increase in the prediction error with propagation distance, from 6 to 8.5 %. Variations of   along 𝑥 are larger for case B.1, maybe because 𝑇 𝑡𝑜𝑡 represents a smaller number of 𝑇 𝑝 for this case. Extending the total recording period could smooth the results but would increase the influence of parasitic reflection.

As a complementary information on the prediction's quality, the correlation indicators separate the error in amplitude and the error in phase. Results are presented in Figure 15.

The match is overall very good between the prediction and the measurements for each probe located in the theoretical prediction area, with a correlation factor 𝐶 𝑚𝑎𝑥 greater than 0.9 in most cases. For case A.10 and (𝑥 -𝑥 0 )∕𝜆 𝑝 ≈ 8.4, the very high and dubious value of 𝐶 𝑚𝑎𝑥 is probably due to the small theoretical prediction duration at this location (see prediction zone in Figure 12a), limiting the relevance of the statistical metrics.

The phase shift remains very close to zero for both sea states and at all studied locations. The low steepness of the sea states tested (𝜖 𝑐 = 1%) was not likely to induce large errors in the wave speeds, even if we were using a linear wave model, so this result was expected. But it is worth checking, as a correct phase information is an essential aspect of a good deterministic wave prediction, especially when aiming to develop control systems.

Figure 10 and 11a are representative examples of a high correlation and a small time shift. A weaker correlation associated with a near-zero time shift, as in Figure 11b, is often due to the presence of short waves in the prediction area (around 𝑡∕𝑇 𝑝 = 5), or to small waves superimposed on larger ones (around 𝑡∕𝑇 𝑝 = -2), which are not reproduced correctly by the prediction simulations. The main waves, on the other hand, remain fairly well resolved. A high correlation associated with a non-zero phase shift translates into a small offset in the arrival time of the wave crest, visible for some large waves around 𝑡∕𝑇 𝑝 = 1 and 4 in Figure 11c.

Discussion

As shown in Figure 13, Figure 10 and Figure 11, the 5% threshold initially selected to define the frequency bounds, and thus the prediction zone, seems too conservative. These figures show possible extensions of the cut-off frequencies with thresholds up to 75% of the energy peak. The spectra used in the experiments were Pierson-Moskowitz spectra, which are rather sharp and whose energy is concentrated around the peak, especially at low frequencies. Therefore the value of 𝑓 𝑚𝑖𝑛 can only be increased carefully, at the risk of missing high-energy components. However, even with the highest threshold tested (which led to 𝑓 𝑚𝑖𝑛 = 86%𝑓 𝑝 ), the prediction quality remains reasonable. The spectrum is not as sharp for 𝑓 > 𝑓 𝑝 , so the effects of extending the cut-off frequencies are more visible in the lowest part of the prediction zone i.e., at the start of the (reconstruction) -prediction. The largest alternative prediction zone, shown in Figure 13, still allows a rather good agreement with the correlation pattern. This only means that a shorter measurement time is needed to obtain a meaningful reconstruction of the sea state, which is of less interest in the context of wave prediction.

Comparing the experimental results presented in Section 5.2 with the literature is not straightforward, as prediction approaches can be very different from one another. As an example, in the work of [START_REF] Desmars | Experimental and numerical assessment of deterministic nonlinear ocean waves prediction algorithms using non-uniformly sampled wave gauges[END_REF], information on the incident waves is collected over a spatio-temporal area rather than at a single point, and prediction errors are calculated at locations relatively close to the measurement area.

In Figure 14, our results show errors in the same order of magnitude as in their Figure 15, but with slightly greater values: in their work, 𝜀  (𝑥) varies from 3.5 to 6.4% depending on the measurement probe and the sea state considered.

The most obvious cause for greater errors in our case is the imperfect quality of the boundary condition fed into the nonlinear propagation model, due to noisy and incomplete velocity measurements in the water column. The postprocessing step was not able to completely abolish this difficulty: the consequences were already visible in Section 5.1, with a boundary condition of poorer quality than when reconstructed from synthetic data.

A second reason is the presence of reflection in the physical wave tank. Indeed, the prediction method considered here uses a velocity measurement device upstream of the prediction area. It can thus only consider the propagation of waves towards increasing 𝑥. In the tank, despite the absorbing beach, reflected waves are propagating towards decreasing 𝑥 (the reflection coefficient has been estimated at approximately 10 -12% of amplitude for regular waves, see Section 4.1). These reflected waves cross the prediction area before being measured by the sensors around 𝑥 0 , which prevents from taking them into account. As the prediction simulations are compared to the measurements from 𝑡 ≥ 𝑇 𝑟𝑎𝑚𝑝 + 𝑥 0 ∕𝐶 𝑔 𝑚𝑖𝑛 , reflection due to the 𝑓 𝑝 frequency component, which is the main contribution, is present from the start.

As a side note, the retrieved horizontal velocity profile was evaluated on a restricted time span in Section 2.4, to limit the influence of reflection. This way, only the measurement capabilities of the sensors and the relevance of the reconstruction method were assessed. With reflection being more and more present over time in the physical wave tank, the reconstruction error calculated in Section 2.4 is not a true indicator of the overall fidelity of the boundary condition really passed to the wave prediction model.

The results presented in this work showed that, provided adequate experimental conditions are met, the proposed prediction method can produce good quality predictions of the sea surface elevation, up to a few peak periods ahead, for moderate sea states characterized by different 𝜆 𝑝 ∕ℎ ratios. In Section 5.1, the error on the reconstructed horizontal velocity profile was greater in the upper water column for case A (𝑇 𝑝 = 2 s). But the averaged error indicators showed a similar and satisfactory quality of prediction for both sea states.

These results were obtained using a wave model based on a HOS method. This allows to enjoy the accuracy of fully nonlinear models, while maintaining an efficiency that is still compatible with real-time prediction, in the long run. The method investigated in this work achieved results of comparable quality to previous attempts using HOS methods, with a shorter computation time because only one forward HOS simulation is needed here. In the present work, the computations were conducted with a single processor and no code optimization, and the CPU time was comparable to the simulation time. Considering we were in a model scale tank configuration with a Froude scaling, this means that the prediction is actually obtained faster than real-time at full scale. These are promising results, compared to the existing non-linear approaches.

Conclusions

Following on from previous numerical studies [START_REF] Huchet | Nonlinear deterministic sea wave prediction using instantaneous velocity profiles[END_REF], this paper reported on the experimental validation of a nonlinear, deterministic wave prediction method, based on wave-induced instantaneous velocity measurements. The method was tested for long-crested, irregular waves with a low steepness, as an experimental proof of concept. The experiments were conducted in a towing tank using wave probes, a single-point ADV velocity sensor and two ADCP velocity profilers, the latter instruments being originally designed to be deployed at sea. A dedicated method combined the velocity sensors measurements to retrieve the instantaneous, horizontal velocity profile, which was not directly accessible using conventional equipment. This profile was used as a forcing field in the nonlinear wave propagation model HOS-NWTpred, to generate and propagate the corresponding sea state, and to produce a prediction of the free surface elevation. The prediction obtained was compared with the measurements of the wave probes. We evaluated the overall feasibility of the prediction method in two parts: first, we assessed the quality of the instantaneous horizontal velocity profiles retrievable from the acoustic instruments; then, we examined the quality of the surface elevation prediction obtained using this information on the sea state.

This study was conducted under challenging conditions due to the use of ADCPs in a lab environment with fresh, clear water. Because of this, the quality of the velocity data measured was not consistent across all trials.

Still, this work describes the first experimental proof of concept of a new prediction algorithm, and gave promising results. When good measuring conditions were ensured, the reconstruction step proved able to provide a satisfying horizontal velocity profile, using imperfect velocity data from the upper part of the water column only. From this reconstructed profile, the wave model generated good quality predictions of free surface elevation. In particular, the phase error on the predicted waves remained very small even several peak wavelengths downstream the measurements location. This allows a longer prediction horizon and is also an important aspect when developing control strategies for wave energy converters or floating offshore wind turbines.

These results are promising for practical applications, as the method allows the use of nonlinear propagation while getting rid of the time-consuming assimilation procedure usually needed to initialize such nonlinear models. This, combined with the efficiency of an HOS propagation model, significantly reduces the CPU time needed to get a prediction, compared to other nonlinear prediction algorithms.

Further work should include the extension of the method to multi-directional wave prediction. Even with several instruments deployed, the data collected will not provide a full knowledge of the multi-directional sea state and some assumptions on its characteristics will be needed. There is also room for improvement in using more effectively the information already available from the measuring devices: in the present study, only the central beam of the ADCPs was used. The slanted beams could be a valuable complementary source of information as well, provided a suitable processing method is developed to handle and combine the different sources of information.

Figure 1 :

 1 Figure 1: Proposed method for deterministic wave prediction.

  (a) Example of a wave spectrum and associated 𝑓 𝑚𝑖𝑛 and 𝑓 𝑚𝑎𝑥 , bounding the retained contributions. Area theoretically accessible to deterministic wave prediction, depending on the measurement duration 𝑇 and on the group speed considered.

Figure 2 :

 2 Figure 2: Definition of the prediction zone in time and space, for a single fixed point measurement.

Figure 3 :

 3 Figure 3: Reconstruction of the horizontal velocity profile 𝑈 (𝑥 0 , 𝑧, 𝑡) from data measured as two vertical velocity profiles 𝑊 (𝑥 0 ± Δ𝑥∕2, 𝑧, 𝑡) and a horizontal velocity single-point measurement 𝑈 (𝑥 0 , 𝑧 𝑟𝑒𝑓 , 𝑡).

Figure 4 :

 4 Figure 4: Schematic of the experimental set-up. Real distances are not respected. Coloured zones in Figure 4b represent the range of ADCP measurements.

Figure 6 :

 6 Figure 6: Normalized noise spectra.

Figure 7 :

 7 Figure 7: Overview of raw ADCP data and associated correlation for one trial.

  Case A.10 (𝑇 𝑝 = 2 s).

  Case B.1 (𝑇 𝑝 = 3 s).

Figure 8 :

 8 Figure 8: Error profiles for 𝑈 𝑟𝑒𝑐 , computed from one single realization for each sea state. ( ) 𝑊 and 𝑈 𝑟𝑒𝑓 from synthetic, reference data. ( ) 𝑊 from experimental data, 𝑈 𝑟𝑒𝑓 from synthetic data. ( ) 𝑊 and 𝑈 𝑟𝑒𝑓 from experimental data.

  Case A.10 (𝑇 𝑝 = 2 s).

  Case B.1 (𝑇 𝑝 = 3 s).

Figure 9 :

 9 Figure 9: Time series of reference (𝑈 𝑛𝑢𝑚 ) and reconstructed (𝑈 𝑟𝑒𝑐 ) horizontal velocities at 𝑧 = 0.

Figure 10 :

 10 Figure 10: Case A.10. Comparison of representative 𝜂 𝑒𝑥𝑝𝑒 (𝑡) and 𝜂 𝑝𝑟𝑒𝑑 (𝑡) time series at 𝑥 𝑝 = 48 m, i.e. 4.5 𝜆 𝑝 downstream of 𝑥 0 . At selected location, the equivalent of   (𝑥) averaged on one sample only is 7.2%. Corresponding 𝐶 𝑚𝑎𝑥 = 96% and 𝑇 𝑠 ∕𝑇 𝑝 = 0. Vertical grey lines () and ( ) outside the prediction zone indicate prediction horizons for alternative frequency bounds selected at respectively 25% and 75% of the energy peak, instead of 5%.

Figure 11 :

 11 Figure 11: Case B.1. Comparison of representative 𝜂 𝑒𝑥𝑝𝑒 (𝑡) and 𝜂 𝑝𝑟𝑒𝑑 (𝑡) time series at three locations in the prediction area. Vertical grey lines () and ( ) outside the prediction zone indicate prediction horizons for alternative frequency bounds selected at respectively 25% and 75% of the energy peak, instead of 5%.

Figure 12 :

 12 Figure 12: Map of (𝑥, 𝑡) for the two trials considered. Dashed vertical lines show the position of the probes. Solid lines indicate the prediction zone for a reconstruction period of 30 𝑇 𝑝 .Frequencies 𝑓 𝑚𝑖𝑛 and 𝑓 𝑚𝑎𝑥 are associated to the extreme group velocities bounding the prediction zone: they are chosen so that outside these frequency bounds, each component's spectral density is less than 5% of the wave spectrum's peak density.

Figure 13 :

 13 Figure 13: Map of 𝑐(𝑥, 𝑡) for the two trials considered. Black solid lines indicate the prediction zone for a reconstruction period of 30 𝑇 𝑝 . Outside the limit frequency bounds, each component's spectral density is less than 5% of the wave spectrum's peak density. Dashed lines () and ( ) show alternative prediction zones for frequency bounds selected using density thresholds of respectively 25% and 75% of the peak density.

Figure 14 :Figure 15 :

 1415 Figure 14: Prediction error for cases A.10 and B.1. Here, 𝑇 𝑡𝑜𝑡 = 57 𝑇 𝑝 for A.10, 𝑇 𝑡𝑜𝑡 = 50 𝑇 𝑝 for B.1 and Δ𝑡 = 1 s.

The maximum sampling rate is 16 Hz for the most recent instruments, but connection and hardware constraints induced some limitations in the present case.

This represents |𝑧 𝑟𝑒𝑓 |∕𝜆 𝑝 ≈ 9% (resp. ≈ 4%) of a peak wavelength for cases A (resp. for cases B), which means velocity orbitals have not yet faded away.

As verified in dedicated tests, this increase is already visible just below 𝑧 = 𝑧 𝑚𝑎𝑥 because the vertical fitting step mentioned in Section A.3 induced small modifications of the profile in the shaded area.

Acknowledgments

This work was supported by the French Agence de la transition écologique (ADEME) and by DynamOcean.

CRediT authorship contribution statement

Marion Huchet: Conceptualization, methodology, software, validation, formal analysis, investigation, data curation, writing -original draft, writing -review and editing, visualization. Aurélien Babarit: Conceptualization, methodology, validation, writing -review and editing, supervision, project administration, funding acquisition. Guillaume Ducrozet: Conceptualization, methodology, software, validation, data curation, writing -review and editing, supervision. Pierre Ferrant: Conceptualization, methodology, validation, writing -review and editing, supervision. Jean-Christophe Gilloteaux: Conceptualization, methodology, validation, writing -review and editing, supervision. Eloi Droniou: Validation, writing -review and editing, supervision, funding acquisition.

Declaration of competing interest

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

A. ADCP data quality control

A.1. Correlation test

The first processing step rules out data points associated to a bad correlation value. Correlation is a quantity provided directly by the ADCPs for each velocity measurement (i.e., for each cell and each ping)and is a proxy for signal quality. Here, after a close examination of the data set, the minimum acceptable correlation is set to 80%: all velocities with a correlation below this threshold are discarded.

This steps leaves very few valid data points in the lower part of the water column. Therefore, velocity measurements are kept only for cells above a custom depth limit. This limit is defined for each trial as the depth below which more than 50% of the data do not pass the correlation test. It generally falls between 𝑧 𝑙𝑖𝑚 = -1.2 m and -0.8 m.

While allowing to identify many erroneous values, the correlation check still leaves out numerous outliers. The most obvious ones are then detected iteratively and independently for each depth cell, with the following criterion: 𝑊 (𝑧, 𝑡) is an outlier if |𝑊 (𝑧, 𝑡) -⟨𝑊 (𝑧)⟩| > 6𝜎 𝑊 (𝑧), where ⟨𝑊 (𝑧)⟩ is the mean value taken by 𝑊 (𝑧) in the considered sample (updated at each iteration).

A.2. Phase-Space Thresholding Method

The resulting velocity data is a partially masked array with missing data in place of outliers. However, in the following steps a complete time array is needed at each cell, so the missing data are filled using 2D cubic interpolation. An iterative Phase-Space Thresholding Method [START_REF] Goring | Despiking Acoustic Doppler Velocimeter Data[END_REF][START_REF] Mori | Noise of Acoustic Doppler Velocimeter Data in Bubbly Flows[END_REF] then detects the remaining outliers, and replaces them using cubic interpolation. This step is conducted independently for each depth cell and requires knowledge of the full time series at each cell.

A.3. Vertical fitting

Finally, after outliers have been identified and replaced, a vertical fitting is applied on the resulting 𝑊 arrays, using a 3-component approximation of the linear expression for 𝑊 in the Airy wave theory:

At each time step, the parameters (𝑎 1 , 𝑘 1 , 𝑎 2 , 𝑘 2 , 𝑎 3 , 𝑘 3 ) are determined using a non-linear least squares method, to find the function that best fits the post-processed experimental 𝑊 profile between 𝑧 𝑙𝑖𝑚 and 𝑧 𝑚𝑎𝑥 . After this last step, the measurements of both ADCPs present themselves in the form of checked and processed 𝑊 fields, over a depth range restricted to [ 𝑧 𝑙𝑖𝑚 ; 𝑧 𝑚𝑎𝑥 ] .